A156363 A triangle sequence related to the Eulerian numbers of the second kind: t(n,m) = Sum_{i=0..m}(-1)^(m-i)*binomial(n-i-1, m-i)*Stirling2(n+i+1, i+1).

1, 1, 3, 1, 6, 25, 1, 13, 76, 350, 1, 28, 242, 1430, 6951, 1, 59, 783, 6023, 35659, 179487, 1, 122, 2527, 25782, 187092, 1108128, 5715424, 1, 249, 8070, 110960, 995595, 6963711, 41250694, 216627840, 1, 504, 25456, 476626, 5337322, 44302510, 302087532, 1789534102, 9528822303

Offset

0,3

Comments

Row sums are: {1, 4, 32, 440, 8652, 222012, 7039076, 265957120, 11670586356, 583472429540, 32744436653656,...}

Links

G. C. Greubel, Rows n=0..100 of triangle, flattened

L. Smiley, Completion of a Rational Function Sequence of Carlitz, page 3.

Formula

t(n,m) = Sum_{i=0..m}(-1)^(m-i)*binomial(n-i-1, m-i)*Stirling2(n+i+1, i+1).

Example

Triangle begins as:
1;
1,   3;
1,   6,    25;
1,  13,    76,    350;
1,  28,   242,   1430,   6951;
1,  59,   783,   6023,  35659,  179487;
1, 122,  2527,  25782, 187092, 1108128,  5715424;
1, 249,  8070, 110960, 995595, 6963711, 41250694, 216627840;

Mathematica

t[n_, m_] = Sum[(-1)^(m-i)*Binomial[n-i-1, m-i]*StirlingS2[n+i+1, i+1], {i, 0, m}]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]//Flatten

Prog

(PARI) {t(n, m) = sum(j=0, m, (-1)^(m-j)*binomial(n-j-1, m-j)*stirling(n+j +1, j+1, 2))};
for(n=0, 10, for(m=0, n, print1(t(n, m), ", "))) \\ G. C. Greubel, Feb 24 2019
(Magma) [[(&+[(-1)^(m-j)*Binomial(n-j-1, m-j)*StirlingSecond(n+j+1, j+1): j in [0..m]]): m in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 24 2019
(Sage) [[sum((-1)^(m-j)*binomial(n-j-1, m-j)*stirling_number2(n+j+1, j+1) for j in (0..m)) for m in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 24 2019

Crossrefs

Cf. A048993 (Stirling2), A008277, A156139, A156364.

Sequence in context: A271969 A351106 A175291 * A221929 A283432 A157866

Adjacent sequences: A156360 A156361 A156362 * A156364 A156365 A156366

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 08 2009

Status

approved