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URL: https://oeis.org/A176418

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A176418
Triangle read by rows: T(n,k) = A022167(n,k)*n!/binomial(n,k) + 1 - n!.
3
1, 1, 1, 1, 3, 1, 1, 21, 21, 1, 1, 217, 497, 217, 1, 1, 2785, 14401, 14401, 2785, 1, 1, 42961, 527809, 1218961, 527809, 42961, 1, 1, 781921, 23866081, 133305985, 133305985, 23866081, 781921, 1, 1, 16490881, 1290574081, 18068561281, 43725975553, 18068561281, 1290574081, 16490881, 1
OFFSET
0,5
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
FORMULA
q=3;
c(n,q)=Product[1 - q^i, {i, 1, n}];
t(n,m,q)=1 - n! + n!*c(n, q)/(c[m, q)*c(n - m, q))/Binomial[n, m]
EXAMPLE
Triangle begins:
1;
1, 1;
1, 3, 1;
1, 21, 21, 1;
1, 217, 497, 217, 1;
1, 2785, 14401, 14401, 2785, 1;
1, 42961, 527809, 1218961, 527809, 42961, 1;
1, 781921, 23866081, 133305985, 133305985, 23866081, 781921, 1;
...
MATHEMATICA
c[n_, q_] = Product[1 - q^i, {i, 1, n}];
t[n_, m_, q_] = 1 - n! + n!*c[n, q]/(c[m, q]*c[n - m, q])/Binomial[n, m];
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
PROG
(PARI) T(n, q=3) = my(c=matpascal(n, q)); vector(n+1, n, vector(n, k, c[n, k]*(n-1)!/binomial(n-1, k-1) + 1 - (n-1)!));
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Nov 19 2025
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Apr 17 2010
EXTENSIONS
Edited by Andrew Howroyd, Nov 19 2025
STATUS
approved