A158198 Triangle T(n, k) = Sum_{j=0..k-1} (-1)^j*binomial(k, j+1)*(k-j)^(n-k), read by rows.

1, 1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 15, 16, 5, 1, 1, 31, 58, 25, 6, 1, 1, 63, 196, 125, 36, 7, 1, 1, 127, 634, 601, 216, 49, 8, 1, 1, 255, 1996, 2765, 1296, 343, 64, 9, 1, 1, 511, 6178, 12265, 7656, 2401, 512, 81, 10, 1, 1, 1023, 18916, 52925, 44136, 16807, 4096, 729, 100, 11, 1

Offset

1,5

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Formula

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

Example

Triangle begins
  1;
  1,   1;
  1,   3,    1;
  1,   7,    4,    1;
  1,  15,   16,    5,    1;
  1,  31,   58,   25,    6,   1;
  1,  63,  196,  125,   36,   7,  1;
  1, 127,  634,  601,  216,  49,  8, 1;
  1, 255, 1996, 2765, 1296, 343, 64, 9, 1;
a(8,4) = 1*4*4^4 - 1*6*3^4 + 1*4*2^4 - 1*1*1^4 = 1024 - 486 + 64 - 1 = 601.

Mathematica

Table[Sum[(-1)^j*Binomial[k, j+1]*(k-j)^(n-k), {j, 0, k-1}], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jun 26 2021 *)_

Prog

(PARI) tabl(nn) = {for (n=1, nn, for (k=1, n, print1(sum(i=0, k-1, (-1)^i*binomial(k, i+1)*(k-i)^(n-k)), ", "); ); print(); ); } \\ Michel Marcus, Jun 19 2013
(Magma)
[(&+[(-1)^j*Binomial(k, j+1)*(k-j)^(n-k): j in [0..k-1]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Jun 26 2021
(Sage)
flatten([[ sum( (-1)^j*binomial(k, j+1)*(k-j)^(n-k) for j in (0..k-1)) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jun 26 2021

Crossrefs

Column 2 is A000225, column 3 is A168583. - Michel Marcus, Jun 19 2013

Sequence in context: A121300 A283595 A128119 * A158793 A112996 A205099

Adjacent sequences: A158195 A158196 A158197 * A158199 A158200 A158201

Keyword

nonn,tabl

Author

Thomas J Engelsma (tom(AT)opertech.com), Mar 13 2009

Status

approved