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A208510 Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.

%I A208510 #94 Sep 04 2021 20:19:46
%S A208510 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 A208510 36,55,50,27,8,1,1,15,49,91,105,77,35,9,1,1,17,64,140,196,182,112,44,
%U A208510 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 A208510 Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.
%C A208510 Row sums:  A083329
%C A208510 Alternating row sums:  1,0,-1,-1,-1,-1,-1,-1,-1,-1,...
%C A208510 Antidiagonal sums: A000071 (-1+Fibonacci numbers)
%C A208510 col 1:  A000012
%C A208510 col 2:  A005408
%C A208510 col 3:  A000290
%C A208510 col 4:  A000330
%C A208510 col 5:  A002415
%C A208510 col 6:  A005585
%C A208510 col 7:  A040977
%C A208510 col 8:  A050486
%C A208510 col 9:  A053347
%C A208510 col 10:  A054333
%C A208510 col 11:  A054334
%C A208510 col 12:  A057788
%C A208510 col 2n-1 of A208510 is column n of A208508
%C A208510 col 2n of A208510 is column n of A208509.
%C A208510 ...
%C A208510 GENERAL DISCUSSION:
%C A208510 A208510 typifies arrays generated by paired recurrence equations of the following form:
%C A208510 u(n,x)=a(n,x)*u(n-1,x)+b(n,x)*v(n-1,x)+c(n,x)
%C A208510 v(n,x)=d(n,x)*u(n-1,x)+e(n,x)*v(n-1,x)+f(n,x).
%C A208510 ...
%C A208510 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 A208510 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 A208510 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 A208510 ...
%C A208510 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 A208510 A: 1,-1,1,-1,1,-1,1.... A033999
%C A208510 B: 1,2,4,8,16,32,64,... powers of 2
%C A208510 C: 1,1,1,1,1,1,1,1,.... A000012
%C A208510 D: 2,2,2,2,2,2,2,2,.... A007395
%C A208510 E: 2,4,6,8,10,12,14,... even numbers
%C A208510 F: 1,1,2,3,5,8,13,21,.. Fibonacci numbers
%C A208510 N: 1,2,3,4,5,6,7,8,.... A000027
%C A208510 O: 1,3,5,7,9,11,13,.... odd numbers
%C A208510 P: 1,3,9,27,81,243,.... powers of 3
%C A208510 S: 1,4,9,16,25,36,49,.. squares
%C A208510 T: 1,3,6,10,15,21,38,.. triangular numbers
%C A208510 Z: 1,0,0,0,0,0,0,0,0,.. A000007
%C A208510 *: (eventually) periodic alternating row sums
%C A208510 ^: has a limiting row; i.e., the polynomials "approach" a power series
%C A208510 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 A208510 ......... a....b....c....d....e....f....code
%C A208510 A034839 u 1....1....0....1....x....0....CCOT
%C A208510 A034867 v 1....1....0....1....x....0....CEN
%C A208510 A210221 u 1....1....0....1....2x...0....BBFF
%C A208510 A210596 v 1....1....0....1....2x...0....BBFF
%C A208510 A105070 v 1....2x...0....1....1....0....BN
%C A208510 A207605 u 1....1....0....1....x+1..0....BCFFN
%C A208510 A106195 v 1....1....0....1....x+1..0....BCFFN
%C A208510 A207606 u 1....1....0....x....x+1..0....DNT
%C A208510 A207607 v 1....1....0....x....x+1..0....DNT
%C A208510 A207608 u 1....1....0....2x...x+1..0....N
%C A208510 A207609 v 1....1....0....2x...x+1..0....C
%C A208510 A207610 u 1....1....0....1....x....1....CF
%C A208510 A207611 v 1....1....0....1....x....1....BCF
%C A208510 A207612 u 1....1....0....1....2x...1....BF
%C A208510 A207613 v 1....1....0....1....2x...1....BF
%C A208510 A207614 u 1....1....0....1....x+1..1....CN
%C A208510 A207615 v 1....1....0....1....x+1..1....CFN
%C A208510 A207616 u 1....1....0....x....1....1....CE
%C A208510 A207617 v 1....1....0....x....1....1....CNO
%C A208510 A029638 u 1....1....0....x....x....1....CDNO
%C A208510 A029635 v 1....1....0....x....x....1....CDNOZ
%C A208510 A207618 u 1....1....0....x....2x...1....N
%C A208510 A207619 v 1....1....0....x....2x...1....CFN
%C A208510 A207620 u 1....1....0....x....x+1..1....DET
%C A208510 A207621 v 1....1....0....x....x+1..1....DNO
%C A208510 A207622 u 1....1....0....2x...1....1....BT
%C A208510 A207623 v 1....1....0....2x...1....1....BN
%C A208510 A207624 u 1....1....0....2x...x....1....N
%C A208510 A102662 v 1....1....0....2x...x....1....CO
%C A208510 A207625 u 1....1....0....2x...x+1..1....T
%C A208510 A207626 v 1....1....0....2x...x+1..1....N
%C A208510 A207627 u 1....1....0....2x...2x...1....BN
%C A208510 A207628 v 1....1....0....2x...2x...1....BCE
%C A208510 A207629 u 1....1....0....x+1..1....1....CET
%C A208510 A207630 v 1....1....0....x+1..1....1....CO
%C A208510 A207631 u 1....1....0....x+1..x....1....DF
%C A208510 A207632 v 1....1....0....x+1..x....1....DEF
%C A208510 A207633 u 1....1....0....x+1..2x...1....F
%C A208510 A207634 v 1....1....0....x+1..2x...1....F
%C A208510 A207635 u 1....1....0....x+1..x+1..1....DN
%C A208510 A207636 v 1....1....0....x+1..x+1..1....CD
%C A208510 A160232 u 1....x....0....1....2x...0....BCFN
%C A208510 A208341 v 1....x....0....1....2x...0....BCFFN
%C A208510 A085478 u 1....x....0....1....x+1..0....CCOFT*
%C A208510 A078812 v 1....x....0....1....x+1..0....CEFN*
%C A208510 A208342 u 1....x....0....x....x....0....CCFNO
%C A208510 A208343 v 1....x....0....x....x....0....BBCDFZ
%C A208510 A208344 u 1....x....0....x....2x...0....CCFN
%C A208510 A208345 v 1....x....0....x....2x...0....CFZ
%C A208510 A094436 u 1....x....0....x....x+1..0....CFFN
%C A208510 A094437 v 1....x....0....x....x+1..0....CEFF
%C A208510 A117919 u 1....x....0....2x...1....0....BCNT
%C A208510 A135837 v 1....x....0....2x...1....0....BCET
%C A208510 A208328 u 1....x....0....2x...x....0....CCOP
%C A208510 A208329 v 1....x....0....2x...x....0....DPZ
%C A208510 A208330 u 1....x....0....2x...x+1..0....CNPT
%C A208510 A208331 v 1....x....0....2x...x+1..0....CN
%C A208510 A208332 u 1....x....0....2x...2x...0....CCE
%C A208510 A208333 v 1....x....0....2x...2x...0....DZ
%C A208510 A208334 u 1....x....0....x+1..1....0....CCNT
%C A208510 A208335 v 1....x....0....x+1..1....0....CCN*
%C A208510 A208336 u 1....x....0....x+1..x....0....CFNT*
%C A208510 A208337 v 1....x....0....x+1..x....0....ACFN*
%C A208510 A208338 u 1....x....0....x+1..2x...0....CNP
%C A208510 A208339 v 1....x....0....x+1..2x...0....BCNP
%C A208510 A202390 u 1....x....0....x+1..x+1..0....CFPTZ*
%C A208510 A208340 v 1....x....0....x+1..x+1..0....FNPZ*
%C A208510 A208508 u 1....x....0....1....1....1....CCES
%C A208510 A208509 v 1....x....0....1....1....1....BCO
%C A208510 A208510 u 1....x....0....1....x....1....CCCNOS*
%C A208510 A029653 v 1....x....0....1....x....1....BCDOSZ*
%C A208510 A208511 u 1....x....0....1....2x...1....BCFO
%C A208510 A208512 v 1....x....0....1....2x...1....BDFO
%C A208510 A208513 u 1....x....0....1....x+1..1....CCES*
%C A208510 A111125 v 1....x....0....1....x+1..1....COO*
%C A208510 A133567 u 1....x....0....x....1....1....CCOTT
%C A208510 A133084 v 1....x....0....x....1....1....BBCEN
%C A208510 A208514 u 1....x....0....x....x....1....CEFN
%C A208510 A208515 v 1....x....0....x....x....1....BCDFN
%C A208510 A208516 u 1....x....0....x....2x...1....CNN
%C A208510 A208517 v 1....x....0....x....2x...1....CCN
%C A208510 A208518 u 1....x....0....x....x+1..1....CFNT
%C A208510 A208519 v 1....x....0....x....x+1..1....NFFT
%C A208510 A208520 u 1....x....0....2x...1....1....BCTT
%C A208510 A208521 v 1....x....0....2x...1....1....BEN
%C A208510 A208522 u 1....x....0....2x...x....1....CCN
%C A208510 A208523 v 1....x....0....2x...x....1....CCO
%C A208510 A208524 u 1....x....0....2x...x+1..1....CT*
%C A208510 A208525 v 1....x....0....2x...x+1..1....ACNP*
%C A208510 A208526 u 1....x....0....2x...2x...1....CEN
%C A208510 A208527 v 1....x....0....2x...2x...1....CCE
%C A208510 A208606 u 1....x....0....x+1..1....1....CCS
%C A208510 A208607 v 1....x....0....x+1..1....1....CNO
%C A208510 A208608 u 1....x....0....x+1..x....1....CFOT
%C A208510 A208609 v 1....x....0....x+1..x....1....DEN*
%C A208510 A208610 u 1....x....0....x+1..2x...1....CO
%C A208510 A208611 v 1....x....0....x+1..2x...1....DE
%C A208510 A208612 u 1....x....0....x+1..x+1..1....CFNS
%C A208510 A208613 v 1....x....0....x+1..x+1..1....CFN*
%C A208510 A105070 u 1....2x...0....1....1....0....BN
%C A208510 A207536 u 1....2x...0....1....1....0....BCT
%C A208510 A208751 u 1....2x...0....1....x+1..0....CDPT
%C A208510 A208752 v 1....2x...0....1....x+1..0....CNP
%C A208510 A135837 u 1....2x...0....x....1....0....BCNT
%C A208510 A117919 v 1....2x...0....x....1....0....BCNT
%C A208510 A208755 u 1....2x...0....x....x....0....BCDEP
%C A208510 A208756 v 1....2x...0....x....x....0....BCCOZ
%C A208510 A208757 u 1....2x...0....x....2x...0....CDEP
%C A208510 A208758 v 1....2x...0....x....2x...0....CCEPZ
%C A208510 A208763 u 1....2x...0....2x...x....0....CDOP
%C A208510 A208764 v 1....2x...0....2x...x....0....CCCP
%C A208510 A208765 u 1....2x...0....2x...x+1..0....CE
%C A208510 A208766 v 1....2x...0....2x...x+1..0....CC
%C A208510 A208747 u 1....2x...0....2x...2x...0....CDE
%C A208510 A208748 v 1....2x...0....2x...2x...0....CCZ
%C A208510 A208749 u 1....2x...0....x+1..1....0....BCOPT
%C A208510 A208750 v 1....2x...0....x+1..1....0....BCNP*
%C A208510 A208759 u 1....2x...0....x+1..2x....0...CE
%C A208510 A208760 v 1....2x...0....x+1..2x....0...BCO
%C A208510 A208761 u 1....2x...0....x+1..x+1...0...BCCT*
%C A208510 A208762 v 1....2x...0....x+1..x+1...0...BNZ*
%C A208510 A208753 u 1....2x...0....1....1.....1...BCS
%C A208510 A208754 v 1....2x...0....1....1.....1...BO
%C A208510 A105045 u 1....2x...0....1....2x....1...BCCOS*
%C A208510 A208659 v 1....2x...0....1....2x....1...BDOSZ*
%C A208510 A208660 u 1....2x...0....1....x+1...1...CDS
%C A208510 A208904 v 1....2x...0....1....x+1...1...CNO
%C A208510 A208905 u 1....2x...0....x....1.....1...BCT
%C A208510 A208906 v 1....2x...0....x....1.....1...BNN
%C A208510 A208907 u 1....2x...0....x....x.....1...BCN
%C A208510 A208756 v 1....2x...0....x....x.....1...BCCE
%C A208510 A208755 u 1....2x...0....x....2x....1...CEN
%C A208510 A208910 v 1....2x...0....x....2x....1...CCE
%C A208510 A208911 u 1....2x...0....x....x+1...1...BCT
%C A208510 A208912 v 1....2x...0....x....x+1...1...BNT
%C A208510 A208913 u 1....2x...0....2x...1.....1...BCT
%C A208510 A208914 v 1....2x...0....2x...1.....1...BEN
%C A208510 A208915 u 1....2x...0....2x...x.....1...CE
%C A208510 A208916 v 1....2x...0....2x...x.....1...CCO
%C A208510 A208919 u 1....2x...0....2x...x+1...1...CT
%C A208510 A208920 v 1....2x...0....2x...x+1...1...N
%C A208510 A208917 u 1....2x...0....2x...2x....1...CEN
%C A208510 A208918 v 1....2x...0....2x...2x....1...CCNP
%C A208510 A208921 u 1....2x...0....x+1..1.....1...BC
%C A208510 A208922 v 1....2x...0....x+1..1.....1...BON
%C A208510 A208923 u 1....2x...0....x+1..x.....1...BCNO
%C A208510 A208908 v 1....2x...0....x+1..x.....1...BDN*
%C A208510 A208909 u 1....2x...0....x+1..2x....1...BN
%C A208510 A208930 v 1....2x...0....x+1..2x....1...DN
%C A208510 A208931 u 1....2x...0....x+1..x+1...1...BCOS
%C A208510 A208932 v 1....2x...0....x+1..x+1...1...BCO*
%C A208510 A207537 u 1....x+1..0....1....1.....0...BCO
%C A208510 A207538 v 1....x+1..0....1....1.....0...BCE
%C A208510 A122075 u 1....x+1..0....1....x.....0...CCFN*
%C A208510 A037027 v 1....x+1..0....1....x.....0...CCFN*
%C A208510 A209125 u 1....x+1..0....1....2x....0...BCFN*
%C A208510 A164975 v 1....x+1..0....1....2x....0...BF
%C A208510 A209126 u 1....x+1..0....x....x.....0...CDFO*
%C A208510 A209127 v 1....x+1..0....x....x.....0...DFOZ*
%C A208510 A209128 u 1....x+1..0....x....2x....0...CDE*
%C A208510 A209129 v 1....x+1..0....x....2x....0...DEZ
%C A208510 A102756 u 1....x+1..0....x....x+1...0...CFNP*
%C A208510 A209130 v 1....x+1..0....x....x+1...0...CCFNP*
%C A208510 A209131 u 1....x+1..0....2x...x.....0...CDEP*
%C A208510 A209132 v 1....x+1..0....2x...x.....0...CNPZ*
%C A208510 A209133 u 1....x+1..0....2x...2x....0...CDN
%C A208510 A209134 v 1....x+1..0....2x...2x....0...CCN*
%C A208510 A209135 u 1....x+1..0....2x...x+1...0...CN*
%C A208510 A209136 v 1....x+1..0....2x...x+1...0...CCS*
%C A208510 A209137 u 1....x+1..0....x+1..x.....0...CFFP*
%C A208510 A209138 v 1....x+1..0....x+1..x.....0...AFFP*
%C A208510 A209139 u 1....x+1..0....x+1..2x....0...CF*
%C A208510 A209140 v 1....x+1..0....x+1..2x....0...BF
%C A208510 A209141 u 1....x+1..0....x+1..x+1...0...BCF*
%C A208510 A209142 v 1....x+1..0....x+1..x+1...0...BFZ*
%C A208510 A209143 u 1....x+1..0....1....1.....1...CCE*
%C A208510 A209144 v 1....x+1..0....1....1.....1...COO*
%C A208510 A209145 u 1....x+1..0....1....x.....1...CCFN*
%C A208510 A122075 v 1....x+1..0....1....x.....1...CCFN*
%C A208510 A209146 u 1....x+1..0....1....2x....1...BCF*
%C A208510 A209147 v 1....x+1..0....1....2x....1...BF
%C A208510 A209148 u 1....x+1..0....1....x+1...1...CCO*
%C A208510 A209149 v 1....x+1..0....1....x+1...1...CDO*
%C A208510 A209150 u 1....x+1..0....x....1.....1...CCNT*
%C A208510 A208335 v 1....x+1..0....x....1.....1...CDNN*
%C A208510 A209151 u 1....x+1..0....x....x.....1...CFN*
%C A208510 A208337 v 1....x+1..0....x....x.....1...ACFN*
%C A208510 A209152 u 1....x+1..0....x....2x....1...CN*
%C A208510 A208339 v 1....x+1..0....x....x.....1...BCN
%C A208510 A209153 u 1....x+1..0....x....x+1...1...CFT*
%C A208510 A208340 v 1....x+1..0....x....x.....1...FNZ*
%C A208510 A209154 u 1....x+1..0....2x...1.....1...BCT*
%C A208510 A209157 v 1....x+1..0....2x...1.....1...BNN
%C A208510 A209158 u 1....x+1..0....2x...x.....1...CN*
%C A208510 A209159 v 1....x+1..0....2x...x.....1...CO*
%C A208510 A209160 u 1....x+1..0....2x...2x....1...CN*
%C A208510 A209161 v 1....x+1..0....2x...2x....1...CE
%C A208510 A209162 u 1....x+1..0....2x...x+1...1...CT*
%C A208510 A209163 v 1....x+1..0....2x...x+1...1...CO*
%C A208510 A209164 u 1....x+1..0....x+1..1.....1...CC*
%C A208510 A209165 v 1....x+1..0....x+1..1.....1...CCN
%C A208510 A209166 u 1....x+1..0....x+1..x.....1...CFF*
%C A208510 A209167 v 1....x+1..0....x+1..x.....1...FF*
%C A208510 A209168 u 1....x+1..0....x+1..2x....1...CF*
%C A208510 A209169 v 1....x+1..0....x+1..2x....1...CF
%C A208510 A209170 u 1....x+1..0....x+1..x+1...1...CF*
%C A208510 A209171 v 1....x+1..0....x+1..x+1...1...CF*
%C A208510 A053538 u x....1....0....1....1.....0...BBCCFN
%C A208510 A076791 v x....1....0....1....1.....0...BBCDF
%C A208510 A209172 u x....1....0....1....2x....0...BCCFF
%C A208510 A209413 v x....1....0....1....2x....0...BCCFF
%C A208510 A094441 u x....1....0....1....x+1...0...CFFFN
%C A208510 A094442 v x....1....0....1....x+1...0...CEFFF
%C A208510 A054142 u x....1....0....x....x+1...0...CCFOT*
%C A208510 A172431 v x....1....0....x....x+1...0...CEFN*
%C A208510 A008288 u x....1....0....2x...1.....0...CCOO*
%C A208510 A035607 v x....1....0....2x...1.....0...ACDE*
%C A208510 A209414 u x....1....0....2x...x+1...0...CCS
%C A208510 A112351 v x....1....0....2x...x+1...0...CON
%C A208510 A209415 u x....1....0....x+1..x.....0...CCTN
%C A208510 A209416 v x....1....0....x+1..x.....0...ACN*
%C A208510 A209417 u x....1....0....x+1..2x....0...CC
%C A208510 A209418 v x....1....0....x+1..2x....0...BBC
%C A208510 A209419 u x....1....0....x+1..x+1...0...CFTZ*
%C A208510 A209420 v x....1....0....x+1..x+1...0...FNZ*
%C A208510 A209421 u x....1....0....1....1.....1...CCN
%C A208510 A209422 v x....1....0....1....1.....1...CD
%C A208510 A209555 u x....1....0....1....x.....1...CNN
%C A208510 A209556 v x....1....0....1....x.....1...CNN
%C A208510 A209557 u x....1....0....1....2x....1...BCN
%C A208510 A209558 v x....1....0....1....2x....1...BN
%C A208510 A209559 u x....1....0....1....x+1...1...CN
%C A208510 A209560 v x....1....0....1....x+1...1...CN
%C A208510 A209561 u x....1....0....x....1.....1...CCNNT*
%C A208510 A209562 v x....1....0....x....1.....1...CDNNT*
%C A208510 A209563 u x....1....0....x....x.....1...CCFT^
%C A208510 A209564 v x....1....0....x....x.....1...CFN^
%C A208510 A209565 u x....1....0....x....2x....1...CC^
%C A208510 A209566 v x....1....0....x....2x....1...BC^
%C A208510 A209567 u x....1....0....x....x+1...1...CNT*
%C A208510 A209568 v x....1....0....x....x+1...1...NNS*
%C A208510 A209569 u x....1....0....2x...1.....1...CNO*
%C A208510 A209570 v x....1....0....2x...1.....1...DNN*
%C A208510 A209571 u x....1....0....2x...x.....1...CCS^
%C A208510 A209572 v x....1....0....2x...x.....1...CN^
%C A208510 A209573 u x....1....0....2x...x+1...1...CNS
%C A208510 A209574 v x....1....0....2x...x+1...1...NO
%C A208510 A209575 u x....1....0....2x...2x....1...CC
%C A208510 A209576 v x....1....0....2x...2x....1...C
%C A208510 A209577 u x....1....0....x+1..1.....1...CNNT
%C A208510 A209578 v x....1....0....x+1..1.....1...CNN
%C A208510 A209579 u x....1....0....x+1..x.....1...CNNT
%C A208510 A209580 v x....1....0....x+1..x.....1...NN*
%C A208510 A209581 u x....1....0....x+1..2x....1...CN
%C A208510 A209582 v x....1....0....x+1..2x....1...BN
%C A208510 A209583 u x....1....0....x+1..x+1...1...CT*
%C A208510 A209584 v x....1....0....x+1..x+1...1...CN*
%C A208510 A121462 u x....x....0....x....x+1...0...BCFFNZ
%C A208510 A208341 v x....x....0....x....x+1...0...BCFFN
%C A208510 A209687 u x....x....0....2x...x+1...0...BCNZ
%C A208510 A208339 v x....x....0....2x...x+1...0...BCN
%C A208510 A115241 u x....x....0....1....1.....1...CDNZ*
%C A208510 A209688 v x....x....0....1....1.....1...DDN*
%C A208510 A209689 u x....x....0....1....x.....1...FNZ^
%C A208510 A209690 v x....x....0....1....x.....1...FN^
%C A208510 A209691 u x....x....0....1....2x....1...BCZ^
%C A208510 A209692 v x....x....0....1....2x....1...BCC^
%C A208510 A209693 u x....x....0....1....x+1...1...NNZ*
%C A208510 A209694 v x....x....0....1....x+1...1...CN*
%C A208510 A209697 u x....x....0....x....x+1...1...BNZ
%C A208510 A209698 v x....x....0....x....x+1...1...BNT
%C A208510 A209699 u x....x....0....2x...1.....1...BNNZ
%C A208510 A209700 v x....x....0....2x...1.....1...BDN
%C A208510 A209701 u x....x....0....2x...x+1...1...NZ
%C A208510 A209702 v x....x....0....2x...x+1...1...N
%C A208510 A209703 u x....x....0....x+1..1.....1...FNTZ
%C A208510 A209704 v x....x....0....x+1..1.....1...FNNT
%C A208510 A209705 u x....x....0....x+1..x+1...1...BNZ*
%C A208510 A209706 v x....x....0....x+1..x+1...1...BCN*
%C A208510 A209695 u x....x+1..0....2x...x+1...0...ACN*
%C A208510 A209696 v x....x+1..0....2x...x+1...0...CDN*
%C A208510 A209830 u x....x+1..0....x+1..2x....0...ACF
%C A208510 A209831 v x....x+1..0....x+1..2x....0...BCF*
%C A208510 A209745 u x....x+1..0....x+1..x+1...0...ABF*
%C A208510 A209746 v x....x+1..0....x+1..x+1...0...BFZ*
%C A208510 A209747 u x....x+1..0....1....1.....1...ADE*
%C A208510 A209748 v x....x+1..0....1....1.....1...DEO
%C A208510 A209749 u x....x+1..0....1....x.....1...ANN*
%C A208510 A209750 v x....x+1..0....1....x.....1...CNO
%C A208510 A209751 u x....x+1..0....1....2x....1...ABN*
%C A208510 A209752 v x....x+1..0....1....2x....1...BN
%C A208510 A209753 u x....x+1..0....1....x+1...1...AN*
%C A208510 A209754 v x....x+1..0....1....x+1...1...NT*
%C A208510 A209755 u x....x+1..0....x....1.....1...AFN
%C A208510 A209756 v x....x+1..0....x....1.....1...FNO*
%C A208510 A209759 u x....x+1..0....x....2x....1...ACF^
%C A208510 A209760 v x....x+1..0....x....2x....1...CF^*
%C A208510 A209761 u x....x+1..0....x.....x+1..1...ABNS*
%C A208510 A209762 v x....x+1..0....x.....x+1..1...BNS*
%C A208510 A209763 u x....x+1..0....2x....1....1...ABN*
%C A208510 A209764 v x....x+1..0....2x....1....1...BNN
%C A208510 A209765 u x....x+1..0....2x....x....1...ACF^*
%C A208510 A209766 v x....x+1..0....2x....x....1...CF^
%C A208510 A209767 u x....x+1..0....2x....x+1..1...AN*
%C A208510 A209768 v x....x+1..0....2x....x+1..1...N*
%C A208510 A209769 u x....x+1..0....x+1...1....1...AF*
%C A208510 A209770 v x....x+1..0....x+1...1....1...FN
%C A208510 A209771 u x....x+1..0....x+1...x....1...ABN*
%C A208510 A209772 v x....x+1..0....x+1...x....1...BN*
%C A208510 A209773 u x....x+1..0....x+1...2x...1...AF
%C A208510 A209774 v x....x+1..0....x+1...2x...1...FN*
%C A208510 A209775 u x....x+1..0....x+1...x+1..1...AB*
%C A208510 A209776 v x....x+1..0....x+1...x+1..1...BC*
%C A208510 A210033 u 1....1....1....1.....x....1...BCN
%C A208510 A210034 v 1....1....1....1.....x....1...BCDFN
%C A208510 A210035 u 1....1....1....1.....2x...1...BBF
%C A208510 A210036 v 1....1....1....1.....2x...1...BBFF
%C A208510 A210037 u 1....1....1....1.....x+1..1...BCFFN
%C A208510 A210038 v 1....1....1....1.....x+1..1...BCFFN
%C A208510 A210039 u 1....1....1....x.....1....1...BCOT
%C A208510 A210040 v 1....1....1....x.....1....1...BCEN
%C A208510 A210042 u 1....1....1....x.....x....1...BCDEOT*
%C A208510 A124927 v 1....1....1....x.....x....1...BCDET*
%C A208510 A210041 u 1....1....1....x.....2x...1...BFO
%C A208510 A209758 v 1....1....1....x.....2x...1...BCFO
%C A208510 A210187 u 1....1....1....x.....x+1..1...DTF*
%C A208510 A210188 v 1....1....1....x.....x+1..1...DNF*
%C A208510 A210189 u 1....1....1....2x....1....1...BT
%C A208510 A210190 v 1....1....1....2x....1....1...BN
%C A208510 A210191 u 1....1....1....2x....x....1...CO*
%C A208510 A210192 v 1....1....1....2x....x....1...CCO*
%C A208510 A210193 u 1....1....1....2x....x+1..1...CPT
%C A208510 A210194 v 1....1....1....2x....x+1..1...CN
%C A208510 A210195 u 1....1....1....2x....2x...1...BOPT*
%C A208510 A210196 v 1....1....1....2x....2x...1...BCC*
%C A208510 A210197 u 1....1....1....x+1...1....1...BCOT
%C A208510 A210198 v 1....1....1....x+1...1....1...BCEN
%C A208510 A210199 u 1....1....1....x+1...x....1...DFT
%C A208510 A210200 v 1....1....1....x+1...x....1...DFO*
%C A208510 A210201 u 1....1....1....x+1...2x...1...BFP
%C A208510 A210202 v 1....1....1....x+1...2x...1...BF
%C A208510 A210203 u 1....1....1....x+1...x+1..1...BDOP
%C A208510 A210204 v 1....1....1....x+1...x+1..1...BCDN*
%C A208510 A210211 u x....1....1....1.....2x...1...BCFN
%C A208510 A210212 v x....1....1....1.....2x...1...BFN
%C A208510 A210213 u x....1....1....1.....x+1..1...CFFN
%C A208510 A210214 v x....1....1....1.....x+1..1...CFFO
%C A208510 A210215 u x....1....1....x.....x....1...BCDFT^
%C A208510 A210216 v x....1....1....x.....x....1...BCFO^
%C A208510 A210217 u x....1....1....x.....2x...1...CDF^
%C A208510 A210218 v x....1....1....x.....2x...1...BCF^
%C A208510 A210219 u x....1....1....x.....x+1..1...CNSTF*
%C A208510 A210220 v x....1....1....x.....x+1..1...FNNT*
%C A208510 A104698 u x....1....1....2x......1..1...CENS*
%C A208510 A210220 v x....1....1....2x....x+1..1...DNNT*
%C A208510 A210223 u x....1....1....2x....x....1...CD^
%C A208510 A210224 v x....1....1....2x....x....1...CO^
%C A208510 A210225 u x....1....1....2x....x+1..1...CNP
%C A208510 A210226 v x....1....1....2x....x+1..1...NOT
%C A208510 A210227 u x....1....1....2x....2x...1...CDP^
%C A208510 A210228 v x....1....1....2x....2x...1...C^
%C A208510 A210229 u x....1....1....x+1...1....1...CFNN
%C A208510 A210230 v x....1....1....x+1...1....1...CCN
%C A208510 A210231 u x....1....1....x+1...x....1...CNT
%C A208510 A210232 v x....1....1....x+1...x....1...NN*
%C A208510 A210233 u x....1....1....x+1...2x...1...CNP
%C A208510 A210234 v x....1....1....x+1...2x...1...BN
%C A208510 A210235 u x....1....1....x+1...x+1..1...CCFPT*
%C A208510 A210236 v x....1....1....x+1...x+1..1...CFN*
%C A208510 A124927 u x....x....1....1.....1....1...BCDEET*
%C A208510 A210042 v x....1....1....x+1...x+1..1...BDEOT*
%C A208510 A210216 u x....x....1....1.....x....1...BCFO^
%C A208510 A210215 v x....x....1....1.....x....1...BCDFT^
%C A208510 A210549 u x....x....1....1.....2x...1...BCF^
%C A208510 A210550 v x....x....1....1.....2x...1...BDF^
%C A208510 A172431 u x....x....1....1.....x+1..1...CEFN*
%C A208510 A210551 v x....x....1....1.....x+1..1...CFOT*
%C A208510 A210552 u x....x....1....x.....1....1...BBCFNO
%C A208510 A210553 v x....x....1....x.....1....1...BNNFB
%C A208510 A208341 u x....x....1....x.....x+1..1...BCFFN
%C A208510 A210554 v x....x....1....x.....x+1..1...BNFFT
%C A208510 A210555 u x....x....1....2x....1....1...BCNN
%C A208510 A210556 v x....x....1....2x....1....1...BENP
%C A208510 A210557 u x....x....1....2x....x+1..1...CNP
%C A208510 A210558 v x....x....1....2x....x+1..1...N
%C A208510 A210559 u x....x....1....x+1...1....1...CEF
%C A208510 A210560 v x....x....1....x+1...1....1...OFNS
%C A208510 A210561 u x....x....1....x+1...x....1...BCNP^
%C A208510 A210562 v x....x....1....x+1...x....1...BDP*^
%C A208510 A210563 u x....x....1....x+1...2x...1...CFP^
%C A208510 A210564 v x....x....1....x+1...2x...1...DF^
%C A208510 A013609 u x....x....1....x+1...x+1..1...BCEPT*
%C A208510 A209757 v x....x....1....x+1...x+1..1...BCOS*
%C A208510 A209819 u x....2x...1....x+1...x....1...CFN^
%C A208510 A209820 v x....2x...1....x+1...x....1...DF^
%C A208510 A209996 u x....2x...1....x+1...2x...1...CP^
%C A208510 A209998 v x....2x...1....x+1...2x...1...DP^
%C A208510 A209999 u x....x+1..1....1.....x+1..1...FN*
%C A208510 A210287 v x....x+1..1....1.....x+1..1...CFT*
%C A208510 A210565 u x....x+1..1....x.....1....1...FNT*
%C A208510 A210595 v x....x+1..1....x.....1....1...FNNT
%C A208510 A210598 u x....x+1..1....x+1...2x...1...FN*
%C A208510 A210599 v x....x+1..1....x+1...2x...1...FN
%C A208510 A210600 u x....x+1..1....x+1...x+1..1...BF*
%C A208510 A210601 v x....x+1..1....x+1...x+1..1...BF*
%C A208510 A210597 u 2x...1....1....x+1...1....1...BF
%C A208510 A210601 v 2x...1....1....x+1...1....1...BFN*
%C A208510 A210603 u 2x...1....1....x+1...x+1..1...BF
%C A208510 A210738 v 2x...1....1....x+1...x+1..1...CBF*
%C A208510 A210739 u 2x...x....1....x+1...x....1...CF^
%C A208510 A210740 v 2x...x....1....x+1...x....1...DF*^
%C A208510 A210741 u 2x...x....1....x+1...x+1..1...BCFO
%C A208510 A210742 v 2x...x....1....x+1...x+1..1...CFO*
%C A208510 A210743 u 2x...x+1..1....x+1...1....1...F
%C A208510 A210744 v 2x...x+1..1....x+1...1....1...FN
%C A208510 A210747 u 2x...x+1..1....x+1...x+1..1...FF
%C A208510 A210748 v 2x...x+1..1....x+1...x+1..1...CFF*
%C A208510 A210749 u x+1..1....1....x+1...2x...1...BCF
%C A208510 A210750 v x+1..1....1....x+1...2x...1...BF
%C A208510 A210751 u x+1..x....1....x+1...2x...1...FNT
%C A208510 A210752 v x+1..x....1....x+1...2x...1...FN
%C A208510 A210753 u x+1..x....1....x+1...x+1..1...BNZ*
%C A208510 A210754 v x+1..x....1....x+1...x+1..1...BCT*
%C A208510 A210755 u x+1..2x...1....x+1...x+1..1...N*
%C A208510 A210756 v x+1..2x...1....x+1...x+1..1...CT*
%C A208510 A210789 u 1....x....0....x+2...x-1..0...CFFN
%C A208510 A210790 v 1....x....0....x+2...x-1..0...CEFF
%C A208510 A210791 u 1....x....0....x-1...x+2..0...CFNP
%C A208510 A210792 v 1....x....0....x-1...x+2..0...CF
%C A208510 A210793 u 1....x+1..0....x+2...x-1..0...CFNP
%C A208510 A210794 v 1....x+1..0....x+2...x-1..0...FPP
%C A208510 A210795 u 1....x....1....x+2...x-1..0...FN
%C A208510 A210796 v 1....x....1....x+2...x-1..0...FO
%C A208510 A210797 u 1....x....0....x+2...x-1..1...CF
%C A208510 A210798 v 1....x....0....x+2...x-1..1...F
%C A208510 A210799 u 1....x+1..1....x+2...x-1..0...FN
%C A208510 A210800 v 1....x+1..1....x+2...x-1..0...F
%C A208510 A210801 u 1....x+1..1....x+2...x-1..1...FN
%C A208510 A210802 v 1....x+1..1....x+2...x-1..1...F
%C A208510 A210803 u 1....x....0....x-1...x+3..0...F*
%C A208510 A210804 v 1....x....0....x-1...x+3..0...F*
%C A208510 A210805 u 1....x....0....x+2...x-1.-1...CFFN
%C A208510 A210806 v 1....x....0....x+2...x-1.-1...FF
%C A208510 A210858 u 1....x....0....x+n...x....0...CFT*
%C A208510 A210859 v 1....x....0....x+n...x....0...FN*
%C A208510 A210860 u 1....x+1..0....x+n...x....0...F
%C A208510 A210861 v 1....x+1..0....x+n...x....0...F*
%C A208510 A210862 u 1....x....1....x+n-1.x....0...FN
%C A208510 A210863 v 1....x....1....x+n-1.x....0...FS
%C A208510 A210864 u 1....x....1....x+n...x....0...FN
%C A208510 A210865 v 1....x....1....x+n...x....0...FT
%C A208510 A210866 u 1....x....0....x+n...x...-x...CFT
%C A208510 A210867 v 1....x....0....x+n...x...-x...FN
%C A208510 A210868 u 1....x....0....x+1...x-1..0...BCFN
%C A208510 A210869 v 1....x....0....x+1...x-1..0...BBCFNZ
%C A208510 A210870 u 1....x....0....x+1...x-1..1...CFFN
%C A208510 A210871 v 1....x....0....x+1...x-1..1...CFF
%C A208510 A210872 u x....1...-1....x.....x....1...BDFZ^
%C A208510 A210873 v x....1...-1....x.....x....1...BCFN^
%C A208510 A210876 u x....1....1....x.....x....x...BCCF^
%C A208510 A210877 v x....1....1....x.....x....x...BDFNZ^
%C A208510 A210878 u x....2x...0....x+1...x....1...DFZ^
%C A208510 A210879 v x....2x...0....x+1...x....1...FC*^
%C A208510 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
%F A208510 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208510 v(n,x)=u(n-1,x)+x*v(n-1,x)+1,
%F A208510 where u(1,x)=1, v(1,x)=1.
%F A208510 Also, u(n,x)=(x+1)*u(n-1,x)+x for n>2, with u(n,2)=x+1.
%e A208510 First five rows:
%e A208510 1
%e A208510 1...1
%e A208510 1...3...1
%e A208510 1...5...4...1
%e A208510 1...7...9...5...1
%e A208510 First five polynomials u(n,x):
%e A208510 1
%e A208510 1 + x
%e A208510 1 + 3x + x^2
%e A208510 1 + 5x + 4x^2 + x^3
%e A208510 1 + 7x + 9x^2 + 5x^3 + x^4
%t A208510 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208510 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208510 v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
%t A208510 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208510 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208510 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208510 TableForm[cu]
%t A208510 Flatten[%]   (* A208510 *)
%t A208510 Table[Expand[v[n, x]], {n, 1, z}]
%t A208510 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208510 TableForm[cv]
%t A208510 Flatten[%]   (* A029653 *)
%o A208510 (Python)
%o A208510 from sympy import Poly
%o A208510 from sympy.abc import x
%o A208510 def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)
%o A208510 def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
%o A208510 def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
%o A208510 for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 27 2017
%Y A208510 Cf. A029653, A208508, A208509.
%K A208510 nonn,tabl
%O A208510 1,5
%A A208510 _Clark Kimberling_, Feb 28 2012
%E A208510 Corrected by _Philippe Deléham_, Apr 10 2012
%E A208510 Corrections and additions by _Clark Kimberling_, May 09 2012
%E A208510 Corrections in the overview by _Georg Fischer_, Sep 04 2021