A339209 Triangle read by rows : inverse of triangle A339207.

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

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

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

Cf. A000332, A339207, A339208.

Sequence in context: A339208 A198105 A347683 * A277529 A354133 A060338

Adjacent sequences: A339206 A339207 A339208 * A339210 A339211 A339212

Keyword

sign,tabl

Author

Michel Marcus, Nov 27 2020

Status

approved