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A339209
Triangle read by rows : inverse of triangle A339207.
3
1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 5, 0, 1, 0, -9, 0, 15, 0, 1, 0, 0, -63, 0, 35, 0, 1, 0, 1485, 0, -231, 0, 70, 0, 1, 0, 0, 18685, 0, -567, 0, 126, 0, 1, 0, -844757, 0, 125515, 0, -945, 0, 210, 0, 1, 0, 0, -14862727, 0, 600655, 0, -693, 0, 330, 0, 1
OFFSET
1,13
LINKS
René Gy, Bernoulli-Stirling Numbers, INTEGERS 20 (2020), #A67. See Table 5 p. 16.
FORMULA
From G. C. Greubel, Jul 21 2022: (Start)
T(n, k) = inverse( A339207(n, k) ).
T(n, n) = 1.
T(n, n-3) = A000332(n), for n > = 3. (End)
EXAMPLE
Triangle begins
1;
0, 1;
0, 0, 1;
0, 1, 0, 1;
0, 0, 5, 0, 1;
0, -9, 0, 15, 0, 1;
0, 0, -63, 0, 35, 0, 1;
0, 1485, 0, -231, 0, 70, 0, 1;
0, 0, 18685, 0, -567, 0, 126, 0, 1;
MATHEMATICA
M[n_, k_]:= M[n, k]= If[k==n, 1, Sum[Binomial[k+j-1, j]*n^j*BernoulliB[j]*(-1)^j*StirlingS1[n, k+j], {j, 0, n-k}]];
mat = Table[M[n, k], {n, 0, 25}, {k, 0, 25}];
T:= Inverse[mat];
Table[T[[n, k]], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 21 2022 *)
PROG
(PARI) TT(n, k) = sum(h=0, n-k, bernfrac(h)*binomial(k+h-1, h)*abs(stirling(n, h+k, 1))*n^h); \\ A339207
lista(nn) = {my(m = 1/matrix(nn, nn, n, k, n--; k--; TT(n, k))); for(n=1, nn, for (k=1, n, print1(m[n, k], ", "))); m; }
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Michel Marcus, Nov 27 2020
STATUS
approved