cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A208509 Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.

Original entry on oeis.org

1, 3, 5, 1, 7, 5, 9, 14, 1, 11, 30, 7, 13, 55, 27, 1, 15, 91, 77, 9, 17, 140, 182, 44, 1, 19, 204, 378, 156, 11, 21, 285, 714, 450, 65, 1, 23, 385, 1254, 1122, 275, 13, 25, 506, 2079, 2508, 935, 90, 1, 27, 650, 3289, 5148, 2717, 442, 15, 29, 819, 5005, 9867
Offset: 1

Views

Author

Clark Kimberling, Feb 27 2012

Keywords

Examples

			First five rows:
  1
  3
  5    1
  7    5
  9   14   1
First five polynomials v(n,x):
  1
  3
  5 +   x
  7 +  5x
  9 + 14x + x^2

Crossrefs

Columns 0 to 5: A005408, A000330, A005585, A050486, A054333, A057788.
Row sums, v(n,1): A003948.
Alternating row sums, v(n,-1): A090131.
Cf. A208508.

Programs

  • 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] + 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[%]    (* A208508 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208509 *)

Formula

u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x) + 1, with u(1,x)=1, v(1,x)=1.
Conjecture: T(n,k) = binomial(n-1,2*k+1) + binomial(n,2*k+1). - Knud Werner, Jan 11 2022

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

Original entry on oeis.org

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
Offset: 1

Views

Author

Clark Kimberling, Feb 28 2012

Keywords

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

Examples

			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

Crossrefs

Cf. A029653, A208508, A208509.

Programs

  • 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 *)
  • 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

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.

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

A111125 Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 7, 14, 7, 1, 9, 30, 27, 9, 1, 11, 55, 77, 44, 11, 1, 13, 91, 182, 156, 65, 13, 1, 15, 140, 378, 450, 275, 90, 15, 1, 17, 204, 714, 1122, 935, 442, 119, 17, 1, 19, 285, 1254, 2508, 2717, 1729, 665, 152, 19, 1, 21, 385, 2079, 5148, 7007, 5733, 2940, 952, 189, 21, 1
Offset: 0

Views

Author

N. J. A. Sloane, Oct 16 2005

Keywords

Comments

Riordan array ((1+x)/(1-x)^2, x/(1-x)^2). Row sums are A002878. Diagonal sums are A003945. Inverse is A113187. An interesting factorization is (1/(1-x), x/(1-x))(1+2*x, x*(1+x)). - Paul Barry, Oct 17 2005
Central coefficients of rows with odd numbers of term are A052227.
From Wolfdieter Lang, Jun 26 2011: (Start)
T(k,s) appears as T_s(k) in the Knuth reference, p. 285.
This triangle is related to triangle A156308(n,m), appearing in this reference as U_m(n) on p. 285, by T(k,s) - T(k-1,s) = A156308(k,s), k>=s>=1 (identity on p. 286). T(k,s) = A156308(k+1,s+1) - A156308(k,s+1), k>=s>=0 (identity on p. 286).
(End)
A111125 is jointly generated with A208513 as an array of coefficients of polynomials v(n,x): initially, u(1,x)= v(1,x)= 1; for n>1, u(n,x)= u(n-1,x) +x*(x+1)*v(n-1) and v(n,x)= u(n-1,x) +x*v(n-1,x) +1. See the Mathematica section. The columns of A111125 are identical to those of A208508. Here, however, the alternating row sums are periodic (with period 1,2,1,-1,-2,-1). - Clark Kimberling, Feb 28 2012
This triangle T(k,s) (with signs and columns scaled with powers of 5) appears in the expansion of Fibonacci numbers F=A000045 with multiples of odd numbers as indices in terms of odd powers of F-numbers. See the Jennings reference, p. 108, Theorem 1. Quoted as Lemma 3 in the Ozeki reference given in A111418. The formula is: F_{(2*k+1)*n} = Sum_{s=0..k} ( T(k,s)*(-1)^((k+s)*n)*5^s*F_{n}^(2*s+1) ), k >= 0, n >= 0. - Wolfdieter Lang, Aug 24 2012
From Wolfdieter Lang, Oct 18 2012: (Start)
This triangle T(k,s) appears in the formula x^(2*k+1) - x^(-(2*k+1)) = Sum_{s=0..k} ( T(k,s)*(x-x^(-1))^(2*s+1) ), k>=0. Prove the inverse formula (due to the Riordan property this will suffice) with the binomial theorem. Motivated to look into this by the quoted paper of Wang and Zhang, eq. (1.4).
Alternating row sums are A057079.
The Z-sequence of this Riordan array is A217477, and the A-sequence is (-1)^n*A115141(n). For the notion of A- and Z-sequences for Riordan triangles see a W. Lang link under A006232. (End)
The signed triangle ((-1)^(k-s))*T(k,s) gives the coefficients of (x^2)^s of the polynomials C(2*k+1,x)/x, with C the monic integer Chebyshev T-polynomials whose coefficients are given in A127672 (C is there called R). See the odd numbered rows there. This signed triangle is the Riordan array ((1-x)/(1+x)^2, x/(1+x)^2). Proof by comparing the o.g.f. of the row polynomials where x is replaced by x^2 with the odd part of the bisection of the o.g.f. for C(n,x)/x. - Wolfdieter Lang, Oct 23 2012
From Wolfdieter Lang, Oct 04 2013: (Start)
The signed triangle S(k,s) := ((-1)^(k-s))*T(k,s) (see the preceding comment) is used to express in a (4*(k+1))-gon the length ratio s(4*(k+1)) = 2*sin(Pi/4*(k+1)) = 2*cos((2*k+1)*Pi/(4*(k+1))) of a side/radius as a polynomial in rho(4*(k+1)) = 2*cos(Pi/4*(k+1)), the length ratio (smallest diagonal)/side:
s(4*(k+1)) = Sum_{s=0..k} ( S(k,s)*rho(4*(k+1))^(2*s+1) ).
This is to be computed modulo C(4*(k+1), rho(4*(k+1)) = 0, the minimal polynomial (see A187360) in order to obtain s(4*(k+1)) as an integer in the algebraic number field Q(rho(4*(k+1))) of degree delta(4*(k+1)) (see A055034). Thanks go to Seppo Mustonen for asking me to look into the problem of the square of the total length in a regular n-gon, where this formula is used in the even n case. See A127677 for the formula in the (4*k+2)-gon. (End)
From Wolfdieter Lang, Aug 14 2014: (Start)
The row polynomials for the signed triangle (see the Oct 23 2012 comment above), call them todd(k,x) = Sum_{s=0..k} ( (-1)^(k-s)*T(k,s)*x^s ) = S(k, x-2) - S(k-1, x-2), k >= 0, with the Chebyshev S-polynomials (see their coefficient triangle (A049310) and S(-1, x) = 0), satisfy the recurrence todd(k, x) = (-1)^(k-1)*((x-4)/2)*todd(k-1, 4-x) + ((x-2)/2)*todd(k-1, x), k >= 1, todd(0, x) = 1. From the Aug 03 2014 comment on A130777.
This leads to a recurrence for the signed triangle, call it S(k,s) as in the Oct 04 2013 comment: S(k,s) = (1/2)*(1 + (-1)^(k-s))*S(k-1,s-1) + (2*(s+1)*(-1)^(k-s) - 1)*S(k-1,s) + (1/2)*(-1)^(k-s)*Sum_{j=0..k-s-2} ( binomial(j+s+2,s)*4^(j+2)* S(k-1, s+1+j) ) for k >= s >= 1, and S(k,s) = 0 if k < s and S(k,0) = (-1)^k*(2*k+1). Note that the recurrence derived from the Riordan A-sequence A115141 is similar but has simpler coefficients: S(k,s) = sum(A115141(j)*S(k-1,s-1+j), j=0..k-s), k >= s >=1.
(End)
From Tom Copeland, Nov 07 2015: (Start)
Rephrasing notes here: Append an initial column of zeros, except for a 1 at the top, to A111125 here. Then the partial sums of the columns of this modified entry are contained in A208513. Append an initial row of zeros to A208513. Then the difference of consecutive pairs of rows of the modified A208513 generates the modified A111125. Cf. A034807 and A127677.
For relations among the characteristic polynomials of Cartan matrices of the Coxeter root groups, Chebyshev polynomials, cyclotomic polynomials, and the polynomials of this entry, see Damianou (p. 20 and 21) and Damianou and Evripidou (p. 7).
As suggested by the equations on p. 7 of Damianou and Evripidou, the signed row polynomials of this entry are given by (p(n,x))^2 = (A(2*n+1, x) + 2)/x = (F(2*n+1, (2-x), 1, 0, 0, ... ) + 2)/x = F(2*n+1, -x, 2*x, -3*x, ..., (-1)^n n*x)/x = -F(2*n+1, x, 2*x, 3*x, ..., n*x)/x, where A(n,x) are the polynomials of A127677 and F(n, ...) are the Faber polynomials of A263196. Cf. A127672 and A127677.
(End)
The row polynomials P(k, x) of the signed triangle S(k, s) = ((-1)^(k-s))*T(k, s) are given from the row polynomials R(2*k+1, x) of triangle A127672 by
P(k, x) = R(2*k+1, sqrt(x))/sqrt(x). - Wolfdieter Lang, May 02 2021

Examples

			Triangle T(k,s) begins:
k\s  0    1     2     3     4     5     6    7    8   9 10
0:   1
1:   3    1
2:   5    5     1
3:   7   14     7     1
4:   9   30    27     9     1
5:  11   55    77    44    11     1
6:  13   91   182   156    65    13     1
7:  15  140   378   450   275    90    15    1
8:  17  204   714  1122   935   442   119   17    1
9:  19  285  1254  2508  2717  1729   665  152   19   1
10: 21  385  2079  5148  7007  5733  2940  952  189  21  1
... Extended and reformatted by _Wolfdieter Lang_, Oct 18 2012
Application for Fibonacci numbers F_{(2*k+1)*n}, row k=3:
F_{7*n} = 7*(-1)^(3*n)*F_n + 14*(-1)^(4*n)*5*F_n^3 + 7*(-1)^(5*n)*5^2*F_n^5 + 1*(-1)^(6*n)*5^3*F_n^7, n>=0. - _Wolfdieter Lang_, Aug 24 2012
Example for the  Z- and A-sequence recurrences  of this Riordan triangle: Z = A217477 = [3,-4,12,-40,...]; T(4,0) = 3*7 -4*14 +12*7 -40*1 = 9. A =  [1, 2, -1, 2, -5, 14, ..]; T(5,2) = 1*30 + 2*27 - 1*9 + 2*1= 77. _Wolfdieter Lang_, Oct 18 2012
Example for the (4*(k+1))-gon length ratio s(4*(k+1))(side/radius) as polynomial in the ratio rho(4*(k+1)) ((smallest diagonal)/side): k=0, s(4) = 1*rho(4) = sqrt(2); k=1, s(8) = -3*rho(8) + rho(8)^3 = sqrt(2-sqrt(2)); k=2, s(12) = 5*rho(12) - 5*rho(12)^3 + rho(12)^5, and C(12,x) = x^4 - 4*x^2 + 1, hence rho(12)^5 = 4*rho(12)^3 - rho(12), and s(12) = 4*rho(12) - rho(12)^3 = sqrt(2 - sqrt(3)). - _Wolfdieter Lang_, Oct 04 2013
Example for the recurrence for the signed triangle S(k,s)= ((-1)^(k-s))*T(k,s) (see the Aug 14 2014 comment above):
S(4,1) = 0 + (-2*2 - 1)*S(3,1) - (1/2)*(3*4^2*S(3,2) + 4*4^3*S(3,3)) = - 5*14 - 3*8*(-7) - 128*1 = -30. The recurrence from the Riordan A-sequence A115141 is S(4,1) = -7 -2*14 -(-7) -2*1 = -30. - _Wolfdieter Lang_, Aug 14 2014

Links

Crossrefs

Mirror image of A082985, which see for further references, etc.
Also closely related to triangles in A098599 and A100218.
Cf. A052227, A085478, A208513, A208508, A211957
Cf. A034807, A127672, A127677, A263196.

Programs

  • Magma
    [((2*n+1)/(n+k+1))*Binomial(n+k+1, 2*k+1): k in [0..n], n in [0..12]];  // G. C. Greubel, Feb 01 2022
  • Mathematica
    (* First program *)
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+1)*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[%]  (* A208513 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A111125 *) (* Clark Kimberling, Feb 28 2012 *)
(* Second program *)
T[n_, k_]:= ((2*n+1)/(2*k+1))*Binomial[n+k, 2*k];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 01 2022 *)
  • Sage
    @CachedFunction
def T(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 3*T(n-1,k) if n==1 else 2*T(n-1,k)
    return T(n-1,k-1) - T(n-2,k) - h
A111125 = lambda n,k: (-1)^(n-k)*T(n,k)
for n in (0..9): [A111125(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Formula

T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
From Peter Bala, Apr 30 2012: (Start)
T(n,k) = binomial(n+k,2*k) + 2*binomial(n+k,2*k+1).
The row generating polynomials P(n,x) are a generalization of the Morgan-Voyce polynomials b(n,x) and B(n,x). They satisfy the recurrence equation P(n,x) = (x+2)*P(n-1,x) - P(n-2,x) for n >= 2, with initial conditions P(0,x) = 1, P(1,x) = x+r+1 and with r = 2. The cases r = 0 and r = 1 give the Morgan-Voyce polynomials A085478 and A078812 respectively. Andre-Jeannin has considered the case of general r.
P(n,x) = U(n+1,1+x/2) + U(n,1+x/2), where U(n,x) denotes the Chebyshev polynomial of the second kind - see A053117. P(n,x) = (2/x)*(T(2*n+2,u)-T(2*n,u)), where u = sqrt((x+4)/4) and T(n,x) denotes the Chebyshev polynomial of the first kind - see A053120. P(n,x) = Product_{k = 1..n} ( x + 4*(sin(k*Pi/(2*n+1)))^2 ). P(n,x) = 1/x*(b(n+1,x) - b(n-1,x)) and P(n,x) = 1/x*{(b(2*n+2,x)+1)/b(n+1,x) - (b(2*n,x)+1)/b(n,x)}, where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478. Cf. A211957.
(End)
From Wolfdieter Lang, Oct 18 2012 (Start)
O.g.f. column No. s: ((1+x)/(1-x)^2)*(x/(1-x)^2)^s, s >= 0. (from the Riordan data given in a comment above).
O.g.f. of the row polynomials R(k,x):= Sum_{s=0..k} ( T(k,s)*x^s ), k>=0: (1+z)/(1-(2+x)*z+z^2) (from the Riordan property).
(End)
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 12 2013

Extensions

More terms from Paul Barry, Oct 17 2005
Showing 1-3 of 3 results.

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