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A208510 Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section. 339
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, 36, 55, 50, 27, 8, 1, 1, 15, 49, 91, 105, 77, 35, 9, 1, 1, 17, 64, 140, 196, 182, 112, 44, 10, 1, 1, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 1, 21, 100, 285, 540, 714 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row sums:  A083329

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

Antidiagonal sums: A000071 (-1+Fibonacci numbers)

col 1:  A000012

col 2:  A005408

col 3:  A000290

col 4:  A000330

col 5:  A002415

col 6:  A005585

col 7:  A040977

col 8:  A050486

col 9:  A053347

col 10:  A054333

col 11:  A054334

col 12:  A057788

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

col 2n of A208510 is column n of A208509.

...

GENERAL DISCUSSION:

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

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

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

...

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.

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.

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-ab+f.

...

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:

A: 1,-1,1,-1,1,-1,1.... A033999

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

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

D: 2,2,2,2,2,2,2,2,.... A007395

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

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

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

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

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

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

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

Z: 1,0,0,0,0,0,0,0,0,.. A000007

*: (eventually) periodic alternating row sums

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

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.

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

A034839 u 1....1....0....1....x....0....CCOT

A034867 v 1....1....0....1....x....0....CEN

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

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

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

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

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

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

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

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

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

A207610 u 1....1....0....1....x....1....CF

A207611 v 1....1....0....1....x....1....BCF

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

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

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

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

A207616 u 1....1....0....x....1....1....CE

A207617 v 1....1....0....x....1....1....CNO

A029638 u 1....1....0....x....x....1....CDNO

A207635 v 1....1....0....x....x....1....CDNOZ

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A208342 u 1....x....0....x....x....0....CCFNO

A208343 v 1....x....0....x....x....0....BBCDFZ

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A208508 u 1....x....0....1....1....1....CCES

A208509 v 1....x....0....1....1....1....BCO

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

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

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

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

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

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

A133567 u 1....x....0....x....1....1....CCOTT

A133084 v 1....x....0....x....1....1....BBCEN

A208514 u 1....x....0....x....x....1....CEFN

A208515 v 1....x....0....x....x....1....BCDFN

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A053538 u x....1....0....1....1.....0...BBCCFN

A076791 v x....1....0....1....1.....0...BBCDF

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A209421 u x....1....0....1....1.....1...CCN

A209422 v x....1....0....1....1.....1...CD

A209555 u x....1....0....1....x.....1...CNN

A209556 v x....1....0....1....x.....1...CNN

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A210033 u 1....1....1....1.....x....1...BCN

A210034 v 1....1....1....1.....x....1...BCDFN

A210035 u 1....1....1....1.....2x...1...BBF

A210036 v 1....1....1....1.....2x...1...BBFF

A210037 u 1....1....1....1.....2x...1...BCFFN

A210038 v 1....1....1....1.....x+1..1...BCFFN

A210039 u 1....1....1....x.....1....1...BCOT

A210040 v 1....1....1....x.....1....1...BCEN

A210042 u 1....1....1....x.....x....1...BCDEOT*

A124927 v 1....1....1....x.....x....1...BCDET*

A210041 u 1....1....1....x.....2x...1...BFO

A209758 v 1....1....1....x.....2x...1...BCFO

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

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

A210189 u 1....1....1....2x....1....1...BT

A210190 v 1....1....1....2x....1....1...BN

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

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

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

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

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

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

A210197 u 1....1....1....x+1...1....1...BCOT

A210198 v 1....1....1....x+1...1....1...BCEN

A210199 u 1....1....1....x+1...x....1...DFT

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

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

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

A210203 u 1....1....1....x+1...x+1..1...BDOP

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

A210211 u x....1....1....1.....2x...1...BCFN

A210212 v x....1....1....1.....2x...1...BFN

A210213 u x....1....1....1.....x+1..1...CFFN

A210214 v x....1....1....1.....x+1..1...CFFO

A210215 u x....1....1....x.....x....1...BCDFT^

A210216 v x....1....1....x.....x....1...BCFO^

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

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

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

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

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

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

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

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

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

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

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

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

A210229 u x....1....1....x+1...1....1...CFNN

A210230 v x....1....1....x+1...1....1...CCN

A210231 u x....1....1....x+1...x....1...CNT

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

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

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

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

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

A124927 u x....x....1....1.....1....1...BCDEET*

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

A210216 u x....x....1....1.....x....1...BCFO^

A210215 v x....x....1....1.....x....1...BCDFT^

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

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

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

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

A210552 u x....x....1....x.....1....1...BBCFNO

A210553 v x....x....1....x.....1....1...BNNFB

A208341 u x....x....1....x.....x+1..1...BCFFN

A210554 v x....x....1....x.....x+1..1...BNFFT

A210555 u x....x....1....2x....1....1...BCNN

A210556 v x....x....1....2x....1....1...BENP

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

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

A210559 u x....x....1....x+1...1....1...CEF

A210560 v x....x....1....x+1...1....1...OFNS

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

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

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

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

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

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

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

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

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

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

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

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

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

A210595 v x....x+1..1....x.....1....1...FNNT

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A210871 v 1....x....0....x.....x....1...CFF

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

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

A210876 u x....1....1....x.....x....x...BCCF^

A210877 v x....1....1....x.....x....x...BDFNZ^

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

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

LINKS

Table of n, a(n) for n=1..72.

FORMULA

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

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

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

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

EXAMPLE

First five rows:

1

1...1

1...3...1

1...5...4...1

1...7...9...5...1

First five polynomials u(n,x):

1

1 + x

1 + 3x + x^2

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

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

MATHEMATICA

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

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

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

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

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

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

TableForm[cu]

Flatten[%]   (* A208510 *)

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

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

TableForm[cv]

Flatten[%]   (* A029653 *)

CROSSREFS

Cf. A029653, A208508, A208509.

Sequence in context: A134836 A180955 A191751 * A131767 A026780 A209421

Adjacent sequences:  A208507 A208508 A208509 * A208511 A208512 A208513

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 28 2012

EXTENSIONS

Corrected by Philippe Deléham, Apr 10 2012

Corrections and additions by Clark Kimberling, May 09 2012

STATUS

approved

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Last modified December 20 04:40 EST 2014. Contains 252241 sequences.