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 A208510 Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section. 345
 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, 672, 450, 210, 65, 12, 1 (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-af+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....2x...0....1....1....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 A029635 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 v 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 u 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...2x....1...CEN A208918 v 1....2x...0....2x...2x....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....2x...x.....0...CDEP* A209132 v 1....x+1..0....2x...x.....0...CNPZ* A209133 u 1....x+1..0....2x...2x....0...CDN A209134 v 1....x+1..0....2x...2x....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....2x....1...CN* A208339 v 1....x+1..0....x....x.....1...BCN A209153 u 1....x+1..0....x....x+1...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....x+1...1...CN A209560 v x....1....0....1....x+1...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....x+1...1...CNT* A209568 v x....1....0....x....x+1...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....x+1..1.....1...CNNT A209578 v x....1....0....x+1..1.....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* A209688 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....2x...1.....1...BNNZ A209700 v x....x....0....2x...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.....x+1..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......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-1...x+2..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....0....x+2...x-1..1...CF A210798 v 1....x....0....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....0....x-1...x+3..0...F* A210804 v 1....x....0....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+1...x-1..1...CFF A210872 u x....1...-1....x.....x....1...BDFZ^ A210873 v x....1...-1....x.....x....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*^ Some of these triangles have irregular row lengths, making it difficult to retrieve individual rows/columns/diagonals without actually computing the recurrence. - Georg Fischer, Sep 04 2021 LINKS 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 *) PROG (Python) from sympy import Poly from sympy.abc import x def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1 def a(n): return Poly(u(n, x), x).all_coeffs()[::-1] for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017 CROSSREFS Cf. A029653, A208508, A208509. Sequence in context: A349182 A191751 A285409 * 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 Corrections in the overview by Georg Fischer, Sep 04 2021 STATUS approved

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Last modified February 1 07:16 EST 2023. Contains 359981 sequences. (Running on oeis4.)