A180246 Triangle T(n,k) read by rows: T(n,k) = Sum_{j=0..k} (-1)^j *binomial(n+1,j)*(k+2-j)^n, 0 <= k < n.

2, 4, -3, 8, -5, 4, 16, 1, 11, -5, 32, 51, 46, -14, 6, 64, 281, 337, 22, 22, -7, 128, 1163, 2472, 1121, 176, -27, 8, 256, 4257, 15703, 15493, 4419, 163, 37, -9, 512, 14563, 88354, 155980, 88486, 14398, 622, -44, 10, 1024, 47785, 455357, 1310024, 1310816, 454730, 48170, 848, 56, -11

Offset: 1

Author

Roger L. Bagula, Aug 19 2010

Keywords

• sign
• tabl
• easy

Comments

Row sums are apparently the 2nd column of A156984, 2, 1, 7, 23,...

Generalizes A008292 in the sense that a term "2" is added to the factor that is raised to the n-th power in the sum of the definition. A term "1" would generate A008292, too (up to index shifts).

Examples

			Triangle begins with:
     2;
     4,    -3;
     8,    -5,      4;
    16,     1,     11,      -5;
    32,    51,     46,     -14,       6;
    64,   281,    337,      22,      22,     -7;
   128,  1163,   2472,    1121,     176,    -27,     8;
   256,  4257,  15703,   15493,    4419,    163,    37,  -9;
   512, 14563,  88354,  155980,   88486,  14398,   622, -44, 10;
  1024, 47785, 455357, 1310024, 1310816, 454730, 48170, 848, 56, -11;
  ...
		

References

• B. Harris and C J. Park, A generalization of Eulerian numbers with a probabilistic Application, Statistics and Probability Letters 20 (1994), page 40

Links

• G. C. Greubel, Rows n=1..100 of triangle, flattened

Crossrefs

Cf. A008292, A157011.

Programs

GAP

Flat(List([1..12], n-> List([0..n-1], k-> Sum([0..k], j-> (-1)^j*Binomial(n+1, j)*(k-j+2)^n )))); # G. C. Greubel, Feb 23 2019

Magma

[[(&+[(-1)^j*Binomial(n+1, j)*(k-j+2)^n: j in [0..k]]): k in [0..n-1]]: n in [1..12]]; // G. C. Greubel, Feb 23 2019

Maple

A180246 := proc(n,k) add( (-1)^v*binomial(n+1,v)*(k+2-v)^n,v=0..k) ; end proc: # R. J. Mathar, Jan 29 2011
P := proc(n,x) option remember; if n = 0 then 1 else
  (n*x+2*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
  expand(%) fi end:
A180246 := (n,k) -> coeff(P(n,x),x,k):
seq(print(seq(A180246(n,k),k=0..n-1)),n=0..10);  # Peter Luschny, Mar 07 2014

Mathematica

t[n_, j_, d_]:= Sum[(-1)^v *Binomial[n+1, v](j+d-v)^n, {v, 0, j}];
Table[Flatten[Table[Table[t[n,k,m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
(* This sequence corresponds to m=2 *)
Table[Sum[(-1)^j*Binomial[n+1, j]*(k-j+2)^n, {j,0,k}], {n,1,12}, {k,0,n-1}]//Flatten

PARI

{T(n,k) = sum(j=0,k, (-1)^j*binomial(n+1, j)*(k-j+2)^n)};
for(n=1,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 23 2019

Sage

[[sum((-1)^j*binomial(n+1, j)*(k-j+2)^n for j in (0..k)) for k in (0..n-1)] for n in (1..12)] # G. C. Greubel, Feb 23 2019

Formula

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