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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.

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A341546 a(n) = Sum_{k=0..n} (-1)^(n+k)*A111516(n,k).

Original entry on oeis.org

1, 0, 2, 12, 50, 280, 1442, 7812, 42338, 232176, 1280642, 7103932, 39579410, 221340808, 1241708834, 6984796852, 39382895810, 222512915680, 1259482604546, 7140546372204, 40541480041970, 230480474747640, 1311841695315362, 7474722997813732, 42631911134818850
Offset: 0

Views

Author

Petros Hadjicostas, Feb 14 2021

Keywords

Examples

			a(3) = (-1)^3*(1 - 7 + 12 - 18) = 12.
a(4) = (-1)^4*(1 - 15 + 32 - 56 + 88) = 50.

Crossrefs

Cf. A111516.

Programs

  • PARI
    G(n,k) = if (k==0, 1, sum(j=1,n, binomial(n,j)*binomial(k+j-2,j-1))); \\ A111516
a(n) = sum(k=0, n, (-1)^(n+k)*G(n,k)); \\ Michel Marcus, Feb 14 2021

Formula

O.g.f.: ((-3*x^2 + 5*x + 4)*sqrt(x^2 - 6*x + 1) - 3*x^3 + 2*x^2 + 5*x)/(sqrt(x^2 - 6*x + 1)*(6*x^2 + 10*x + 4)).
a(n) ~ sqrt(9*sqrt(2) - 8) * (1 + sqrt(2))^(2*n) / (14*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 14 2021
D-finite with recurrence -2*(n-1)*(56*n^2-328*n+467)*a(n) +(392*n^3-3024*n^2+7501*n-5979)*a(n-1) +(1400*n^3-10328*n^2+24
187*n-17824)*a(n-2) +(728*n^3-5216*n^2+11919*n-8571)*a(n-3) -3*(n-4)*(56*n^2-216*n+195)*a(n-4)=0. - R. J. Mathar, Mar 06 2022
D-finite with recurrence 10*(-n+1)*a(n) +(31*n-53)*a(n-1) +(145*n-274)*a(n-2) +2*(47*n-69)*a(n-3) +2*(-32*n+179)*a(n-4) +3*(-15*n+77)*a(n-5) +9*(n-6)*a(n-6)=0. - R. J. Mathar, Mar 06 2022

A055807 Triangle T read by rows: T(i,j) = R(i-j,j), where R(i,0) = 1 for i >= 0, R(0,j) = 0 for j >= 1, and R(i,j) = Sum_{h=0..i-1, k=0..j} R(h,k) for i >= 1 and j >= 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 4, 1, 0, 1, 15, 12, 5, 1, 0, 1, 31, 32, 18, 6, 1, 0, 1, 63, 80, 56, 25, 7, 1, 0, 1, 127, 192, 160, 88, 33, 8, 1, 0, 1, 255, 448, 432, 280, 129, 42, 9, 1, 0, 1, 511, 1024, 1120, 832, 450, 180, 52, 10, 1, 0, 1, 1023
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Comments

Formatted as a triangular array, it is [1, 0, 1, 1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 0, -1, 1, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 05 2006
The square array (R(n,k): n,k >= 0) referred to in the name of the sequence is actually A050143. - Petros Hadjicostas, Feb 13 2021

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,   0;
  1,   1,   0;
  1,   3,   1,   0;
  1,   7,   4,   1,   0;
  1,  15,  12,   5,   1,   0;
  1,  31,  32,  18,   6,   1,  0;
  1,  63,  80,  56,  25,   7,  1, 0;
  1, 127, 192, 160,  88,  33,  8, 1, 0;
  1, 255, 448, 432, 280, 129, 42, 9, 1, 0;
  ...
Florez et al. (2019) give the triangle in this form:
    1,    0,    0,   0,   0,   0,  0,  0, 0, ...
    3,    1,    0,   0,   0,   0,  0,  0, 0, ...
    7,    4,    1,   0,   0,   0,  0,  0, 0, ...
   15,   12,    5,   1,   0,   0,  0,  0, 0, ...
   31,   32,   18,   6,   1,   0,  0,  0, 0, ...
   63,   80,   56,  25,   7,   1,  0,  0, 0, ...
  127,  192,  160,  88,  33,   8,  1,  0, 0, ...
  255,  448,  432, 280, 129,  42,  9,  1, 0, ...
  511, 1024, 1120, 832, 450, 180, 52, 10, 1, ...
  ...

Links

Crossrefs

Rows sums: A001519 (odd-indexed Fibonacci numbers).
Cf. A050143, A050147, A050148, A111516.
Cf. A055809, A055810, A055811, A055815, A055816, A055817.

Programs

  • GAP
    T:= function(i,j)
    if j=0 then return 1;
    elif i=0 then return 0;
    else return Sum([0..i-1], h-> Sum([0..j], m-> T(h,m) ));
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n-k,k) ))); # G. C. Greubel, Jan 23 2020
  • Magma
    function T(i,j)
  if j eq 0 then return 1;
  elif i eq 0 then return 0;
  else return (&+[(&+[T(h,m): m in [0..j]]): h in [0..i-1]]);
  end if; return T; end function;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 23 2020
  • Maple
    T:= proc(i, j) option remember;
      if j=0 then 1
    elif i=0 then 0
    else add(add(T(h,m), m=0..j), h=0..i-1)
      fi; end:
seq(seq(T(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Jan 23 2020
  • Mathematica
    T[i_, j_]:= T[i, j]= If[j==0, 1, If[i==0, 0, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]]; Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 23 2020 *)
  • PARI
    T(i,j) = if(j==0, 1, if(i==0, 0, sum(h=0,i-1, sum(m=0,j, T(h,m) ))));
for(n=0,12, for(k=0, n, print1(T(n-k,k), ", "))) \\ G. C. Greubel, Jan 23 2020
  • Sage
    @CachedFunction
def T(i, j):
    if j==0: return 1
    elif i==0: return 0
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[[T(n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 23 2020

Formula

T(2*n,n) = A050146(n).
G.f.: (1-2*x)*(1-x*y)/((1-x)*(1-x*y-2*x+x^2*y)). - R. J. Mathar, Aug 11 2015
From Petros Hadjicostas, Feb 13 2021: (Start)
T(n,k) = A050143(n-k, k) for 0 <= k <= n.
T(n,k) = (n-k)*hypergeom([-n + k + 1, k], [2], -1) = Sum_{s=1..n-k} binomial(n-k,s)*binomial(s+k-2,k-1) for 1 <= k <= n.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 2 <= k <= n-1 with initial conditions T(n,0) = 1 for n >= 0, T(n,n) = 0 for n >= 1, and T(n,1) = 2^(n-1) - 1 for n >= 2. (End)
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