A152260 Triangle T(n, k) = [x^k] p(n, x), where p(n, x) = (1/n)*(1-x)^(2*n) * Sum_{j >= 0} binomial(n+j-1, j) * j^n * x^(j-1).

1, 1, 2, 1, 10, 9, 1, 32, 113, 64, 1, 86, 786, 1526, 625, 1, 212, 4182, 18932, 24337, 7776, 1, 498, 19167, 170332, 477807, 450066, 117649, 1, 1136, 80103, 1266400, 6584615, 12910704, 9492289, 2097152, 1, 2542, 314928, 8313394, 72899230, 254556594, 375886768, 225159022, 43046721

Offset

1,3

Comments

A generalization of for n=m: q(x,n,m) = (1-x)^(n+m) * Sum_{j >= 0} binomial(m+j-1, j) * j^n * x^(j-1).

References

Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91

Links

G. C. Greubel, Table of n, a(n) for n = 1..1275

Formula

T(n, k) = [x^k] p(n, x), where p(n, x) = (1/n)*(1-x)^(2*n) * Sum_{j >= 0} binomial(n+j-1, j) * j^n * x^(j-1).

Sum_{k=1..n} T(n, k) = A006963(n).

T(n, m) = Sum_{k=1..m} binomial(n+k,k)*binomial(n-k,m-k)*k!*(-1)^(m-k) * Stirling2(n,k)*1/(n+k) (conjectured). - Michael D. Weiner, Jul 01 2020

From G. C. Greubel, May 23 2023: (Start)

T(n, k) = (1/n)*Sum_{j=0..k-1} (-1)^(k+j+1)*binomial(2*n, k-j-1) * binomial(n+j, j+1) * (j+1)^n.

T(n, n) = A000169(n).

T(n, n-1) = A176824(n). (End)

Example

Polynomials, p(n, x), begin as:
  p(1, x) = 1;
  p(2, x) = 1 +   2*x;
  p(3, x) = 1 +  10*x +    9*x^2;
  p(4, x) = 1 +  32*x +  113*x^2 +    64*x^3;
  p(5, x) = 1 +  86*x +  786*x^2 +  1526*x^3 +   625*x^4;
  p(6, x) = 1 + 212*x + 4182*x^2 + 18932*x^3 + 24337*x^4 + 7776*x^5;
Triangle, T(n, k), begins as:
  1;
  1,    2;
  1,   10,     9;
  1,   32,   113,      64;
  1,   86,   786,    1526,     625;
  1,  212,  4182,   18932,   24337,     7776;
  1,  498, 19167,  170332,  477807,   450066,  117649;
  1, 1136, 80103, 1266400, 6584615, 12910704, 9492289, 2097152;

Mathematica

p[x_, n_]:= ((1-x)^(2*n)/n)*Sum[Binomial[k+n-1, k]*k^n*x^(k-1), {k, 0, Infinity}];
Table[CoefficientList[p[x, n], x], {n, 12}]//Flatten
(* Second program *)
T[n_, k_]:= (1/n)*Sum[Binomial[2*n, k-j]*Binomial[n+j-1, j]*(-1)^(k+j) *j^n, {j, k}];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, May 23 2023 *)

Prog

(Magma)
A152260:= func< n, k | (&+[(-1)^(k+j)*Binomial(2*n, k-j)*Binomial(n+j-1, j)*j^n: j in [1..k]])/n >;
[A152260(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, May 23 2023
(SageMath)
def A152260(n, k): return (1/n)*sum( (-1)^(k+j)*binomial(2*n, k-j)*binomial(n+j-1, j)*j^n for j in range(1, k+1) )
flatten([[A152260(n, k) for k in range(1, n+1)] for n in range(1, 13)]) # G. C. Greubel, May 23 2023

Crossrefs

Cf. A000169, A006963, A123125, A176824.

Sequence in context: A060694 A217108 A127259 * A286781 A193727 A138098

Adjacent sequences: A152257 A152258 A152259 * A152261 A152262 A152263

Keyword

nonn,tabl

Author

Roger L. Bagula, Dec 01 2008

Status

approved