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

Author

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

Keywords

• nonn
• tabl

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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

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 *)_

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

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

Formula

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