A156535 Triangle, T(n, k, q) = e(n, k, q), where e(n, k, q) = ((1 - (-q)^k)/(1+q))*e(n-1, k, q) + (-q)^(k-1)*e(n-1, k-1, q), e(n, 0, q) = e(n, n, q) = 1, and q = 2, read by rows.

1, 1, 1, 1, -3, 1, 1, 1, -9, 1, 1, -3, -23, 67, 1, 1, 1, -81, -151, 1083, 1, 1, -3, -239, 1403, 9497, -34677, 1, 1, 1, -729, -5103, 126915, 424313, -2219285, 1, 1, -3, -2183, 31347, 1314417, -12971853, -68273223, 284068395, 1, 1, 1, -6561, -139271, 14960139, 230347569, -3765947181, -15406841031, 72721509291, 1

Offset

1,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

R. Parthasarathy, q-Fermionic Numbers and Their Roles in Some Physical Problems, arXiv:quant-ph/0403216, 2004.

Formula

T(n, k, q) = e(n, k, q), where e(n, k, q) = ((1 - (-q)^k)/(1 + q))*e(n-1, k, q) + (-q)^(k-1)*e(n-1, k-1, q), e(n, 0, q) = e(n, n, q) = 1, and q = 2.

Example

Triangle begins as:
  1;
  1,  1;
  1, -3,     1;
  1,  1,    -9,     1;
  1, -3,   -23,    67,       1;
  1,  1,   -81,  -151,    1083,         1;
  1, -3,  -239,  1403,    9497,    -34677,         1;
  1,  1,  -729, -5103,  126915,    424313,  -2219285,         1;
  1, -3, -2183, 31347, 1314417, -12971853, -68273223, 284068395,      1;

Mathematica

e[n_, k_, q_]:= e[n, k, q]= If[k<0 || k>n, 0, If[k==1 || k==n, 1, ((1-(-q)^k)/(1+q))*e[n-1, k, q] + (-q)^(k-1)*e[n-1, k-1, q] ]];
T[n_, k_, q_]:= e[n, k, q];
Table[T[n, k, 2], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Jan 03 2022 *)

Prog

(Sage)
def e(n, k, q):
    if (k<0 or k>n): return 0
    elif (k==1 or k==n): return 1
    else: return ((1-(-q)^k)/(1+q))*e(n-1, k, q) + (-q)^(k-1)*e(n-1, k-1, q)
def A156535(n, k, q): return e(n, k, q)
flatten([[A156535(n, k, 2) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jan 03 2022

Crossrefs

Cf. A156538, A156539.

Sequence in context: A343717 A263159 A229142 * A365427 A366787 A348038

Adjacent sequences: A156532 A156533 A156534 * A156536 A156537 A156538

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 09 2009

Extensions

Edited by G. C. Greubel, Jan 03 2022

Status

approved