login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A208510 Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section. 345

%I

%S 1,1,1,1,3,1,1,5,4,1,1,7,9,5,1,1,9,16,14,6,1,1,11,25,30,20,7,1,1,13,

%T 36,55,50,27,8,1,1,15,49,91,105,77,35,9,1,1,17,64,140,196,182,112,44,

%U 10,1,1,19,81,204,336,378,294,156,54,11,1,1,21,100,285,540,714,672,450,210,65,12,1

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.

%C Row sums: A083329

%C Alternating row sums: 1,0,-1,-1,-1,-1,-1,-1,-1,-1,...

%C Antidiagonal sums: A000071 (-1+Fibonacci numbers)

%C col 1: A000012

%C col 2: A005408

%C col 3: A000290

%C col 4: A000330

%C col 5: A002415

%C col 6: A005585

%C col 7: A040977

%C col 8: A050486

%C col 9: A053347

%C col 10: A054333

%C col 11: A054334

%C col 12: A057788

%C col 2n-1 of A208510 is column n of A208508

%C col 2n of A208510 is column n of A208509.

%C ...

%C GENERAL DISCUSSION:

%C A208510 typifies arrays generated by paired recurrence equations of the following form:

%C u(n,x)=a(n,x)*u(n-1,x)+b(n,x)*v(n-1,x)+c(n,x)

%C v(n,x)=d(n,x)*u(n-1,x)+e(n,x)*v(n-1,x)+f(n,x).

%C ...

%C These first-order recurrences imply separate second-order recurrences. In order to show them, the six functions a(n,x),...,f(n,x) are abbreviated as a,b,c,d,e,f.

%C Then, starting with initial values u(1,x)=1 and u(2,x)=a+b+c: u(n,x) = (a+e)u(n-1,x) + (bd-ae)u(n-2,x) + bf-ce+c.

%C With initial values v(1,x)=1 and v(2,x)=d+e+f: v(n,x) = (a+e)v(n-1,x) + (bd-ae)v(n-2,x) + cd-af+f.

%C ...

%C In the guide below, the last column codes certain sequences that occur in one of these ways: row, column, edge, row sum, alternating row sum. Coding:

%C A: 1,-1,1,-1,1,-1,1.... A033999

%C B: 1,2,4,8,16,32,64,... powers of 2

%C C: 1,1,1,1,1,1,1,1,.... A000012

%C D: 2,2,2,2,2,2,2,2,.... A007395

%C E: 2,4,6,8,10,12,14,... even numbers

%C F: 1,1,2,3,5,8,13,21,.. Fibonacci numbers

%C N: 1,2,3,4,5,6,7,8,.... A000027

%C O: 1,3,5,7,9,11,13,.... odd numbers

%C P: 1,3,9,27,81,243,.... powers of 3

%C S: 1,4,9,16,25,36,49,.. squares

%C T: 1,3,6,10,15,21,38,.. triangular numbers

%C Z: 1,0,0,0,0,0,0,0,0,.. A000007

%C *: (eventually) periodic alternating row sums

%C ^: has a limiting row; i.e., the polynomials "approach" a power series

%C This coding includes indirect and repeated occurrences; e.g. F occurs thrice at A094441: in column 1 directly as Fibonacci numbers, in row sums as odd-indexed Fibonacci numbers, and in alternating row sums as signed Fibonacci numbers.

%C ......... a....b....c....d....e....f....code

%C A034839 u 1....1....0....1....x....0....CCOT

%C A034867 v 1....1....0....1....x....0....CEN

%C A210221 u 1....1....0....1....2x...0....BBFF

%C A210596 v 1....1....0....1....2x...0....BBFF

%C A105070 v 1....1....0....1....2x...0....BN

%C A207605 u 1....1....0....1....x+1..0....BCFFN

%C A106195 v 1....1....0....1....x+1..0....BCFFN

%C A207606 u 1....1....0....x....x+1..0....DNT

%C A207607 v 1....1....0....x....x+1..0....DNT

%C A207608 u 1....1....0....2x...x+1..0....N

%C A207609 v 1....1....0....2x...x+1..0....C

%C A207610 u 1....1....0....1....x....1....CF

%C A207611 v 1....1....0....1....x....1....BCF

%C A207612 u 1....1....0....1....2x...1....BF

%C A207613 v 1....1....0....1....2x...1....BF

%C A207614 u 1....1....0....1....x+1..1....CN

%C A207615 v 1....1....0....1....x+1..1....CFN

%C A207616 u 1....1....0....x....1....1....CE

%C A207617 v 1....1....0....x....1....1....CNO

%C A029638 u 1....1....0....x....x....1....CDNO

%C A207635 v 1....1....0....x....x....1....CDNOZ

%C A207618 u 1....1....0....x....2x...1....N

%C A207619 v 1....1....0....x....2x...1....CFN

%C A207620 u 1....1....0....x....x+1..1....DET

%C A207621 v 1....1....0....x....x+1..1....DNO

%C A207622 u 1....1....0....2x...1....1....BT

%C A207623 v 1....1....0....2x...1....1....BN

%C A207624 u 1....1....0....2x...x....1....N

%C A102662 v 1....1....0....2x...x....1....CO

%C A207625 u 1....1....0....2x...x+1..1....T

%C A207626 v 1....1....0....2x...x+1..1....N

%C A207627 u 1....1....0....2x...2x...1....BN

%C A207628 v 1....1....0....2x...2x...1....BCE

%C A207629 u 1....1....0....x+1..1....1....CET

%C A207630 v 1....1....0....x+1..1....1....CO

%C A207631 u 1....1....0....x+1..x....1....DF

%C A207632 v 1....1....0....x+1..x....1....DEF

%C A207633 u 1....1....0....x+1..2x...1....F

%C A207634 v 1....1....0....x+1..2x...1....F

%C A207635 u 1....1....0....x+1..x+1..1....DN

%C A207636 v 1....1....0....x+1..x+1..1....CD

%C A160232 u 1....x....0....1....2x...0....BCFN

%C A208341 v 1....x....0....1....2x...0....BCFFN

%C A085478 u 1....x....0....1....x+1..0....CCOFT*

%C A078812 v 1....x....0....1....x+1..0....CEFN*

%C A208342 u 1....x....0....x....x....0....CCFNO

%C A208343 v 1....x....0....x....x....0....BBCDFZ

%C A208344 u 1....x....0....x....2x...0....CCFN

%C A208345 v 1....x....0....x....2x...0....CFZ

%C A094436 u 1....x....0....x....x+1..0....CFFN

%C A094437 v 1....x....0....x....x+1..0....CEFF

%C A117919 u 1....x....0....2x...1....0....BCNT

%C A135837 v 1....x....0....2x...1....0....BCET

%C A208328 u 1....x....0....2x...x....0....CCOP

%C A208329 v 1....x....0....2x...x....0....DPZ

%C A208330 u 1....x....0....2x...x+1..0....CNPT

%C A208331 v 1....x....0....2x...x+1..0....CN

%C A208332 u 1....x....0....2x...2x...0....CCE

%C A208333 v 1....x....0....2x...2x...0....DZ

%C A208334 u 1....x....0....x+1..1....0....CCNT

%C A208335 v 1....x....0....x+1..1....0....CCN*

%C A208336 u 1....x....0....x+1..x....0....CFNT*

%C A208337 v 1....x....0....x+1..x....0....ACFN*

%C A208338 u 1....x....0....x+1..2x...0....CNP

%C A208339 v 1....x....0....x+1..2x...0....BCNP

%C A202390 u 1....x....0....x+1..x+1..0....CFPTZ*

%C A208340 v 1....x....0....x+1..x+1..0....FNPZ*

%C A208508 u 1....x....0....1....1....1....CCES

%C A208509 v 1....x....0....1....1....1....BCO

%C A208510 u 1....x....0....1....x....1....CCCNOS*

%C A029653 v 1....x....0....1....x....1....BCDOSZ*

%C A208511 u 1....x....0....1....2x...1....BCFO

%C A208512 v 1....x....0....1....2x...1....BDFO

%C A208513 u 1....x....0....1....x+1..1....CCES*

%C A111125 v 1....x....0....1....x+1..1....COO*

%C A133567 u 1....x....0....x....1....1....CCOTT

%C A133084 v 1....x....0....x....1....1....BBCEN

%C A208514 u 1....x....0....x....x....1....CEFN

%C A208515 v 1....x....0....x....x....1....BCDFN

%C A208516 u 1....x....0....x....2x...1....CNN

%C A208517 v 1....x....0....x....2x...1....CCN

%C A208518 u 1....x....0....x....x+1..1....CFNT

%C A208519 v 1....x....0....x....x+1..1....NFFT

%C A208520 u 1....x....0....2x...1....1....BCTT

%C A208521 u 1....x....0....2x...1....1....BEN

%C A208522 u 1....x....0....2x...x....1....CCN

%C A208523 v 1....x....0....2x...x....1....CCO

%C A208524 u 1....x....0....2x...x+1..1....CT*

%C A208525 v 1....x....0....2x...x+1..1....ACNP*

%C A208526 u 1....x....0....2x...2x...1....CEN

%C A208527 v 1....x....0....2x...2x...1....CCE

%C A208606 u 1....x....0....x+1..1....1....CCS

%C A208607 v 1....x....0....x+1..1....1....CNO

%C A208608 u 1....x....0....x+1..x....1....CFOT

%C A208609 v 1....x....0....x+1..x....1....DEN*

%C A208610 u 1....x....0....x+1..2x...1....CO

%C A208611 v 1....x....0....x+1..2x...1....DE

%C A208612 u 1....x....0....x+1..x+1..1....CFNS

%C A208613 v 1....x....0....x+1..x+1..1....CFN*

%C A105070 u 1....2x...0....1....1....0....BN

%C A207536 v 1....2x...0....1....1....0....BCT

%C A208751 u 1....2x...0....1....x+1..0....CDPT

%C A208752 v 1....2x...0....1....x+1..0....CNP

%C A135837 u 1....2x...0....x....1....0....BCNT

%C A117919 v 1....2x...0....x....1....0....BCNT

%C A208755 u 1....2x...0....x....x....0....BCDEP

%C A208756 v 1....2x...0....x....x....0....BCCOZ

%C A208757 u 1....2x...0....x....2x...0....CDEP

%C A208758 v 1....2x...0....x....2x...0....CCEPZ

%C A208763 u 1....2x...0....2x...x....0....CDOP

%C A208764 v 1....2x...0....2x...x....0....CCCP

%C A208765 u 1....2x...0....2x...x+1..0....CE

%C A208766 v 1....2x...0....2x...x+1..0....CC

%C A208747 u 1....2x...0....2x...2x...0....CDE

%C A208748 v 1....2x...0....2x...2x...0....CCZ

%C A208749 u 1....2x...0....x+1..1....0....BCOPT

%C A208750 v 1....2x...0....x+1..1....0....BCNP*

%C A208759 u 1....2x...0....x+1..2x....0...CE

%C A208760 v 1....2x...0....x+1..2x....0...BCO

%C A208761 u 1....2x...0....x+1..x+1...0...BCCT*

%C A208762 v 1....2x...0....x+1..x+1...0...BNZ*

%C A208753 u 1....2x...0....1....1.....1...BCS

%C A208754 v 1....2x...0....1....1.....1...BO

%C A105045 u 1....2x...0....1....2x....1...BCCOS*

%C A208659 v 1....2x...0....1....2x....1...BDOSZ*

%C A208660 u 1....2x...0....1....x+1...1...CDS

%C A208904 v 1....2x...0....1....x+1...1...CNO

%C A208905 u 1....2x...0....x....1.....1...BCT

%C A208906 v 1....2x...0....x....1.....1...BNN

%C A208907 u 1....2x...0....x....x.....1...BCN

%C A208756 v 1....2x...0....x....x.....1...BCCE

%C A208755 u 1....2x...0....x....2x....1...CEN

%C A208910 v 1....2x...0....x....2x....1...CCE

%C A208911 u 1....2x...0....x....x+1...1...BCT

%C A208912 v 1....2x...0....x....x+1...1...BNT

%C A208913 u 1....2x...0....2x...1.....1...BCT

%C A208914 v 1....2x...0....2x...1.....1...BEN

%C A208915 u 1....2x...0....2x...x.....1...CE

%C A208916 v 1....2x...0....2x...x.....1...CCO

%C A208919 u 1....2x...0....2x...x+1...1...CT

%C A208920 v 1....2x...0....2x...x+1...1...N

%C A208917 u 1....2x...0....2x...2.....1...CEN

%C A208918 v 1....2x...0....2x...2.....1...CCNP

%C A208921 u 1....2x...0....x+1..1.....1...BC

%C A208922 v 1....2x...0....x+1..1.....1...BON

%C A208923 u 1....2x...0....x+1..x.....1...BCNO

%C A208908 v 1....2x...0....x+1..x.....1...BDN*

%C A208909 u 1....2x...0....x+1..2x....1...BN

%C A208930 v 1....2x...0....x+1..2x....1...DN

%C A208931 u 1....2x...0....x+1..x+1...1...BCOS

%C A208932 v 1....2x...0....x+1..x+1...1...BCO*

%C A207537 u 1....x+1..0....1....1.....0...BCO

%C A207538 v 1....x+1..0....1....1.....0...BCE

%C A122075 u 1....x+1..0....1....x.....0...CCFN*

%C A037027 v 1....x+1..0....1....x.....0...CCFN*

%C A209125 u 1....x+1..0....1....2x....0...BCFN*

%C A164975 v 1....x+1..0....1....2x....0...BF

%C A209126 u 1....x+1..0....x....x.....0...CDFO*

%C A209127 v 1....x+1..0....x....x.....0...DFOZ*

%C A209128 u 1....x+1..0....x....2x....0...CDE*

%C A209129 v 1....x+1..0....x....2x....0...DEZ

%C A102756 u 1....x+1..0....x....x+1...0...CFNP*

%C A209130 v 1....x+1..0....x....x+1...0...CCFNP*

%C A209131 u 1....x+1..0....x....x+1...0...CDEP*

%C A209132 v 1....x+1..0....x....x+1...0...CNPZ*

%C A209133 u 1....x+1..0....x....x+1...0...CDN

%C A209134 v 1....x+1..0....x....x+1...0...CCN*

%C A209135 u 1....x+1..0....2x...x+1...0...CN*

%C A209136 v 1....x+1..0....2x...x+1...0...CCS*

%C A209137 u 1....x+1..0....x+1..x.....0...CFFP*

%C A209138 v 1....x+1..0....x+1..x.....0...AFFP*

%C A209139 u 1....x+1..0....x+1..2x....0...CF*

%C A209140 v 1....x+1..0....x+1..2x....0...BF

%C A209141 u 1....x+1..0....x+1..x+1...0...BCF*

%C A209142 v 1....x+1..0....x+1..x+1...0...BFZ*

%C A209143 u 1....x+1..0....1....1.....1...CCE*

%C A209144 v 1....x+1..0....1....1.....1...COO*

%C A209145 u 1....x+1..0....1....x.....1...CCFN*

%C A122075 v 1....x+1..0....1....x.....1...CCFN*

%C A209146 u 1....x+1..0....1....2x....1...BCF*

%C A209147 v 1....x+1..0....1....2x....1...BF

%C A209148 u 1....x+1..0....1....x+1...1...CCO*

%C A209149 v 1....x+1..0....1....x+1...1...CDO*

%C A209150 u 1....x+1..0....x....1.....1...CCNT*

%C A208335 v 1....x+1..0....x....1.....1...CDNN*

%C A209151 u 1....x+1..0....x....x.....1...CFN*

%C A208337 v 1....x+1..0....x....x.....1...ACFN*

%C A209152 u 1....x+1..0....x....x.....1...CN*

%C A208339 v 1....x+1..0....x....x.....1...BCN

%C A209153 u 1....x+1..0....x....x.....1...CFT*

%C A208340 v 1....x+1..0....x....x.....1...FNZ*

%C A209154 u 1....x+1..0....2x...1.....1...BCT*

%C A209157 v 1....x+1..0....2x...1.....1...BNN

%C A209158 u 1....x+1..0....2x...x.....1...CN*

%C A209159 v 1....x+1..0....2x...x.....1...CO*

%C A209160 u 1....x+1..0....2x...2x....1...CN*

%C A209161 v 1....x+1..0....2x...2x....1...CE

%C A209162 u 1....x+1..0....2x...x+1...1...CT*

%C A209163 v 1....x+1..0....2x...x+1...1...CO*

%C A209164 u 1....x+1..0....x+1..1.....1...CC*

%C A209165 v 1....x+1..0....x+1..1.....1...CCN

%C A209166 u 1....x+1..0....x+1..x.....1...CFF*

%C A209167 v 1....x+1..0....x+1..x.....1...FF*

%C A209168 u 1....x+1..0....x+1..2x....1...CF*

%C A209169 v 1....x+1..0....x+1..2x....1...CF

%C A209170 u 1....x+1..0....x+1..x+1...1...CF*

%C A209171 v 1....x+1..0....x+1..x+1...1...CF*

%C A053538 u x....1....0....1....1.....0...BBCCFN

%C A076791 v x....1....0....1....1.....0...BBCDF

%C A209172 u x....1....0....1....2x....0...BCCFF

%C A209413 v x....1....0....1....2x....0...BCCFF

%C A094441 u x....1....0....1....x+1...0...CFFFN

%C A094442 v x....1....0....1....x+1...0...CEFFF

%C A054142 u x....1....0....x....x+1...0...CCFOT*

%C A172431 v x....1....0....x....x+1...0...CEFN*

%C A008288 u x....1....0....2x...1.....0...CCOO*

%C A035607 v x....1....0....2x...1.....0...ACDE*

%C A209414 u x....1....0....2x...x+1...0...CCS

%C A112351 v x....1....0....2x...x+1...0...CON

%C A209415 u x....1....0....x+1..x.....0...CCTN

%C A209416 v x....1....0....x+1..x.....0...ACN*

%C A209417 u x....1....0....x+1..2x....0...CC

%C A209418 v x....1....0....x+1..2x....0...BBC

%C A209419 u x....1....0....x+1..x+1...0...CFTZ*

%C A209420 v x....1....0....x+1..x+1...0...FNZ*

%C A209421 u x....1....0....1....1.....1...CCN

%C A209422 v x....1....0....1....1.....1...CD

%C A209555 u x....1....0....1....x.....1...CNN

%C A209556 v x....1....0....1....x.....1...CNN

%C A209557 u x....1....0....1....2x....1...BCN

%C A209558 v x....1....0....1....2x....1...BN

%C A209559 u x....1....0....1....2x....1...CN

%C A209560 v x....1....0....1....2x....1...CN

%C A209561 u x....1....0....x....1.....1...CCNNT*

%C A209562 v x....1....0....x....1.....1...CDNNT*

%C A209563 u x....1....0....x....x.....1...CCFT^

%C A209564 v x....1....0....x....x.....1...CFN^

%C A209565 u x....1....0....x....2x....1...CC^

%C A209566 v x....1....0....x....2x....1...BC^

%C A209567 u x....1....0....x....2x....1...CNT*

%C A209568 v x....1....0....x....2x....1...NNS*

%C A209569 u x....1....0....2x...1.....1...CNO*

%C A209570 v x....1....0....2x...1.....1...DNN*

%C A209571 u x....1....0....2x...x.....1...CCS^

%C A209572 v x....1....0....2x...x.....1...CN^

%C A209573 u x....1....0....2x...x+1...1...CNS

%C A209574 v x....1....0....2x...x+1...1...NO

%C A209575 u x....1....0....2x...2x....1...CC

%C A209576 v x....1....0....2x...2x....1...C

%C A209577 u x....1....0....2x...2x....1...CNNT

%C A209578 v x....1....0....2x...2x....1...CNN

%C A209579 u x....1....0....x+1..x.....1...CNNT

%C A209580 v x....1....0....x+1..x.....1...NN*

%C A209581 u x....1....0....x+1..2x....1...CN

%C A209582 v x....1....0....x+1..2x....1...BN

%C A209583 u x....1....0....x+1..x+1...1...CT*

%C A209584 v x....1....0....x+1..x+1...1...CN*

%C A121462 u x....x....0....x....x+1...0...BCFFNZ

%C A208341 v x....x....0....x....x+1...0...BCFFN

%C A209687 u x....x....0....2x...x+1...0...BCNZ

%C A208339 v x....x....0....2x...x+1...0...BCN

%C A115241 u x....x....0....1....1.....1...CDNZ*

%C A209668 v x....x....0....1....1.....1...DDN*

%C A209689 u x....x....0....1....x.....1...FNZ^

%C A209690 v x....x....0....1....x.....1...FN^

%C A209691 u x....x....0....1....2x....1...BCZ^

%C A209692 v x....x....0....1....2x....1...BCC^

%C A209693 u x....x....0....1....x+1...1...NNZ*

%C A209694 v x....x....0....1....x+1...1...CN*

%C A209697 u x....x....0....x....x+1...1...BNZ

%C A209698 v x....x....0....x....x+1...1...BNT

%C A209699 u x....x....0....x....x+1...1...BNNZ

%C A209700 v x....x....0....x....x+1...1...BDN

%C A209701 u x....x....0....2x...x+1...1...NZ

%C A209702 v x....x....0....2x...x+1...1...N

%C A209703 u x....x....0....x+1..1.....1...FNTZ

%C A209704 v x....x....0....x+1..1.....1...FNNT

%C A209705 u x....x....0....x+1..x+1...1...BNZ*

%C A209706 v x....x....0....x+1..x+1...1...BCN*

%C A209695 u x....x+1..0....2x...x+1...0...ACN*

%C A209696 v x....x+1..0....2x...x+1...0...CDN*

%C A209830 u x....x+1..0....x+1..2x....0...ACF

%C A209831 v x....x+1..0....x+1..2x....0...BCF*

%C A209745 u x....x+1..0....x+1..x+1...0...ABF*

%C A209746 v x....x+1..0....x+1..x+1...0...BFZ*

%C A209747 u x....x+1..0....1....1.....1...ADE*

%C A209748 v x....x+1..0....1....1.....1...DEO

%C A209749 u x....x+1..0....1....x.....1...ANN*

%C A209750 v x....x+1..0....1....x.....1...CNO

%C A209751 u x....x+1..0....1....2x....1...ABN*

%C A209752 v x....x+1..0....1....2x....1...BN

%C A209753 u x....x+1..0....1....x+1...1...AN*

%C A209754 v x....x+1..0....1....x+1...1...NT*

%C A209755 u x....x+1..0....x....1.....1...AFN

%C A209756 v x....x+1..0....x....1.....1...FNO*

%C A209759 u x....x+1..0....x....2x....1...ACF^

%C A209760 v x....x+1..0....x....2x....1...CF^*

%C A209761 u x....x+1..0....x.....x+1..1...ABNS*

%C A209762 v x....x+1..0....x.....x+1..1...BNS*

%C A209763 u x....x+1..0....2x....1....1...ABN*

%C A209764 v x....x+1..0....2x....1....1...BNN

%C A209765 u x....x+1..0....2x....x....1...ACF^*

%C A209766 v x....x+1..0....2x....x....1...CF^

%C A209767 u x....x+1..0....2x....x+1..1...AN*

%C A209768 v x....x+1..0....2x....x+1..1...N*

%C A209769 u x....x+1..0....x+1...1....1...AF*

%C A209770 v x....x+1..0....x+1...1....1...FN

%C A209771 u x....x+1..0....x+1...x....1...ABN*

%C A209772 v x....x+1..0....x+1...x....1...BN*

%C A209773 u x....x+1..0....x+1...2x...1...AF

%C A209774 v x....x+1..0....x+1...2x...1...FN*

%C A209775 u x....x+1..0....x+1...x+1..1...AB*

%C A209776 v x....x+1..0....x+1...x+1..1...BC*

%C A210033 u 1....1....1....1.....x....1...BCN

%C A210034 v 1....1....1....1.....x....1...BCDFN

%C A210035 u 1....1....1....1.....2x...1...BBF

%C A210036 v 1....1....1....1.....2x...1...BBFF

%C A210037 u 1....1....1....1.....2x...1...BCFFN

%C A210038 v 1....1....1....1.....x+1..1...BCFFN

%C A210039 u 1....1....1....x.....1....1...BCOT

%C A210040 v 1....1....1....x.....1....1...BCEN

%C A210042 u 1....1....1....x.....x....1...BCDEOT*

%C A124927 v 1....1....1....x.....x....1...BCDET*

%C A210041 u 1....1....1....x.....2x...1...BFO

%C A209758 v 1....1....1....x.....2x...1...BCFO

%C A210187 u 1....1....1....x.....x+1..1...DTF*

%C A210188 v 1....1....1....x.....x+1..1...DNF*

%C A210189 u 1....1....1....2x....1....1...BT

%C A210190 v 1....1....1....2x....1....1...BN

%C A210191 u 1....1....1....2x....x....1...CO*

%C A210192 v 1....1....1....2x....x....1...CCO*

%C A210193 u 1....1....1....2x....x+1..1...CPT

%C A210194 v 1....1....1....2x....x+1..1...CN

%C A210195 u 1....1....1....2x....2x...1...BOPT*

%C A210196 v 1....1....1....2x....2x...1...BCC*

%C A210197 u 1....1....1....x+1...1....1...BCOT

%C A210198 v 1....1....1....x+1...1....1...BCEN

%C A210199 u 1....1....1....x+1...x....1...DFT

%C A210200 v 1....1....1....x+1...x....1...DFO*

%C A210201 u 1....1....1....x+1...2x...1...BFP

%C A210202 v 1....1....1....x+1...2x...1...BF

%C A210203 u 1....1....1....x+1...x+1..1...BDOP

%C A210204 v 1....1....1....x+1...x+1..1...BCDN*

%C A210211 u x....1....1....1.....2x...1...BCFN

%C A210212 v x....1....1....1.....2x...1...BFN

%C A210213 u x....1....1....1.....x+1..1...CFFN

%C A210214 v x....1....1....1.....x+1..1...CFFO

%C A210215 u x....1....1....x.....x....1...BCDFT^

%C A210216 v x....1....1....x.....x....1...BCFO^

%C A210217 u x....1....1....x.....2x...1...CDF^

%C A210218 v x....1....1....x.....2x...1...BCF^

%C A210219 u x....1....1....x.....x+1..1...CNSTF*

%C A210220 v x....1....1....x.....x+1..1...FNNT*

%C A104698 u x....1....1....2x....x+1..1...CENS*

%C A210220 v x....1....1....2x....x+1..1...DNNT*

%C A210223 u x....1....1....2x....x....1...CD^

%C A210224 v x....1....1....2x....x....1...CO^

%C A210225 u x....1....1....2x....x+1..1...CNP

%C A210226 v x....1....1....2x....x+1..1...NOT

%C A210227 u x....1....1....2x....2x...1...CDP^

%C A210228 v x....1....1....2x....2x...1...C^

%C A210229 u x....1....1....x+1...1....1...CFNN

%C A210230 v x....1....1....x+1...1....1...CCN

%C A210231 u x....1....1....x+1...x....1...CNT

%C A210232 v x....1....1....x+1...x....1...NN*

%C A210233 u x....1....1....x+1...2x...1...CNP

%C A210234 v x....1....1....x+1...2x...1...BN

%C A210235 u x....1....1....x+1...x+1..1...CCFPT*

%C A210236 v x....1....1....x+1...x+1..1...CFN*

%C A124927 u x....x....1....1.....1....1...BCDEET*

%C A210042 v x....1....1....x+1...x+1..1...BDEOT*

%C A210216 u x....x....1....1.....x....1...BCFO^

%C A210215 v x....x....1....1.....x....1...BCDFT^

%C A210549 u x....x....1....1.....2x...1...BCF^

%C A210550 v x....x....1....1.....2x...1...BDF^

%C A172431 u x....x....1....1.....x+1..1...CEFN*

%C A210551 v x....x....1....1.....x+1..1...CFOT*

%C A210552 u x....x....1....x.....1....1...BBCFNO

%C A210553 v x....x....1....x.....1....1...BNNFB

%C A208341 u x....x....1....x.....x+1..1...BCFFN

%C A210554 v x....x....1....x.....x+1..1...BNFFT

%C A210555 u x....x....1....2x....1....1...BCNN

%C A210556 v x....x....1....2x....1....1...BENP

%C A210557 u x....x....1....2x....x+1..1...CNP

%C A210558 v x....x....1....2x....x+1..1...N

%C A210559 u x....x....1....x+1...1....1...CEF

%C A210560 v x....x....1....x+1...1....1...OFNS

%C A210561 u x....x....1....x+1...x....1...BCNP^

%C A210562 v x....x....1....x+1...x....1...BDP*^

%C A210563 u x....x....1....x+1...2x...1...CFP^

%C A210564 v x....x....1....x+1...2x...1...DF^

%C A013609 u x....x....1....x+1...x+1..1...BCEPT*

%C A209757 v x....x....1....x+1...x+1..1...BCOS*

%C A209819 u x....2x...1....x+1...x....1...CFN^

%C A209820 v x....2x...1....x+1...x....1...DF^

%C A209996 u x....2x...1....x+1...2x...1...CP^

%C A209998 v x....2x...1....x+1...2x...1...DP^

%C A209999 u x....x+1..1....1.....x+1..1...FN*

%C A210287 v x....x+1..1....1.....x+1..1...CFT*

%C A210565 u x....x+1..1....x.....1....1...FNT*

%C A210595 v x....x+1..1....x.....1....1...FNNT

%C A210598 u x....x+1..1....x+1...2x...1...FN*

%C A210599 v x....x+1..1....x+1...2x...1...FN

%C A210600 u x....x+1..1....x+1...x+1..1...BF*

%C A210601 v x....x+1..1....x+1...x+1..1...BF*

%C A210597 u 2x...1....1....x+1...1....1...BF

%C A210601 v 2x...1....1....x+1...1....1...BFN*

%C A210603 u 2x...1....1....x+1...x+1..1...BF

%C A210738 v 2x...1....1....x+1...x+1..1...CBF*

%C A210739 u 2x...x....1....x+1...x....1...CF^

%C A210740 v 2x...x....1....x+1...x....1...DF*^

%C A210741 u 2x...x....1....x+1...x+1..1...BCFO

%C A210742 v 2x...x....1....x+1...x+1..1...CFO*

%C A210743 u 2x...x+1..1....x+1...1....1...F

%C A210744 v 2x...x+1..1....x+1...1....1...FN

%C A210747 u 2x...x+1..1....x+1...x+1..1...FF

%C A210748 v 2x...x+1..1....x+1...x+1..1...CFF*

%C A210749 u x+1..1....1....x+1...2x...1...BCF

%C A210750 v x+1..1....1....x+1...2x...1...BF

%C A210751 u x+1..x....1....x+1...2x...1...FNT

%C A210752 v x+1..x....1....x+1...2x...1...FN

%C A210753 u x+1..x....1....x+1...x+1..1...BNZ*

%C A210754 v x+1..x....1....x+1...x+1..1...BCT*

%C A210755 u x+1..2x...1....x+1...x+1..1...N*

%C A210756 v x+1..2x...1....x+1...x+1..1...CT*

%C A210789 u 1....x....0....x+2...x-1..0...CFFN

%C A210790 v 1....x....0....x+2...x-1..0...CEFF

%C A210791 u 1....x....0....x+2...x-1..0...CFNP

%C A210792 v 1....x....0....x-1...x+2..0...CF

%C A210793 u 1....x+1..0....x+2...x-1..0...CFNP

%C A210794 v 1....x+1..0....x+2...x-1..0...FPP

%C A210795 u 1....x....1....x+2...x-1..0...FN

%C A210796 v 1....x....1....x+2...x-1..0...FO

%C A210797 u 1....x....1....x+2...x-1..1...CF

%C A210798 v 1....x....1....x+2...x-1..1...F

%C A210799 u 1....x+1..1....x+2...x-1..0...FN

%C A210800 v 1....x+1..1....x+2...x-1..0...F

%C A210801 u 1....x+1..1....x+2...x-1..1...FN

%C A210802 v 1....x+1..1....x+2...x-1..1...F

%C A210803 u 1....x....1....x-1...x+3..0...F*

%C A210804 v 1....x....1....x-1...x+3..0...F*

%C A210805 u 1....x....0....x+2...x-1..1...CFFN

%C A210806 v 1....x....0....x+2...x-1..1...FF

%C A210858 u 1....x....0....x+n...x....0...CFT*

%C A210859 v 1....x....0....x+n...x....0...FN*

%C A210860 u 1....x+1..0....x+n...x....0...F

%C A210861 v 1....x+1..0....x+n...x....0...F*

%C A210862 u 1....x....1....x+n-1.x....0...FN

%C A210863 v 1....x....1....x+n-1.x....0...FS

%C A210864 u 1....x....1....x+n...x....0...FN

%C A210865 v 1....x....1....x+n...x....0...FT

%C A210866 u 1....x....0....x+n...x...-x...CFT

%C A210867 v 1....x....0....x+n...x...-x...FN

%C A210868 u 1....x....0....x+1...x-1..0...BCFN

%C A210869 v 1....x....0....x+1...x-1..0...BBCFNZ

%C A210870 u 1....x....0....x+1...x-1..1...CFFN

%C A210871 v 1....x....0....x.....x....1...CFF

%C A210872 u x....1...-1....x.....x....1...BDFZ^

%C A210873 v x....1...-1....x+1...x-1..1...BCFN^

%C A210876 u x....1....1....x.....x....x...BCCF^

%C A210877 v x....1....1....x.....x....x...BDFNZ^

%C A210878 u x....2x...0....x+1...x....1...DFZ^

%C A210879 v x....2x...0....x+1...x....1...FC*^

%F u(n,x)=u(n-1,x)+x*v(n-1,x),

%F v(n,x)=u(n-1,x)+x*v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1.

%F Also, u(n,x)=(x+1)*u(n-1,x)+x for n>2, with u(n,2)=x+1.

%e First five rows:

%e 1

%e 1...1

%e 1...3...1

%e 1...5...4...1

%e 1...7...9...5...1

%e First five polynomials u(n,x):

%e 1

%e 1 + x

%e 1 + 3x + x^2

%e 1 + 5x + 4x^2 + x^3

%e 1 + 7x + 9x^2 + 5x^3 + x^4

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];

%t v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A208510 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A029653 *)

%o (Python)

%o from sympy import Poly

%o def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)

%o def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1

%o def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]

%o for n in xrange(1, 13): print a(n) # _Indranil Ghosh_, May 27 2017

%Y Cf. A029653, A208508, A208509.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Feb 28 2012

%E Corrected by _Philippe Deléham_, Apr 10 2012

%E Corrections and additions by _Clark Kimberling_, May 09 2012

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 20 15:15 EDT 2019. Contains 328267 sequences. (Running on oeis4.)