A156538 Triangle T(n, k, q) = e(n, k, q) + e(n, n-k+1, 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 = 3, read by rows.

2, 2, 2, 2, -10, 2, 2, -31, -31, 2, 2, 989, -406, 989, 2, 2, 81578, -16213, -16213, 81578, 2, 2, -19816168, 3777869, 670556, 3777869, -19816168, 2, 2, -14445938413, 2685823244, 251846999, 251846999, 2685823244, -14445938413, 2

Offset

1,1

Links

G. C. Greubel, Rows n = 1..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) + e(n, n-k+1, 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 = 3.

Example

Triangle begins as:
  2;
  2,            2;
  2,          -10,          2;
  2,          -31,        -31,         2;
  2,          989,       -406,       989,         2;
  2,        81578,     -16213,    -16213,     81578,          2;
  2,    -19816168,    3777869,    670556,   3777869,  -19816168,            2;
  2, -14445938413, 2685823244, 251846999, 251846999, 2685823244, -14445938413, 2;

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] + e[n, n-k+1, q];
Table[T[n, k, 3], {n, 12}, {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 A156538(n, k, q): return e(n, k, q) + e(n, n-k+1, q)
flatten([[A156538(n, k, 3) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jan 03 2022

Crossrefs

Cf. A156535, A156539.

Sequence in context: A085056 A371619 A265447 * A249768 A217503 A165466

Adjacent sequences: A156535 A156536 A156537 * A156539 A156540 A156541

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 09 2009

Extensions

Edited by G. C. Greubel, Jan 03 2022

Status

approved