A003128 -id:A003128 - cp's OEIS frontend

A118984 Triangular T(n,k) which contains in column k >= 0 the elements of the Stirling transform of the unsigned sequence Stirling1(j+k,j), j >= 0.

1, 2, 1, 5, 6, 2, 15, 31, 23, 6, 52, 160, 195, 110, 24, 203, 856, 1505, 1365, 634, 120, 877, 4802, 11312, 14560, 10738, 4284, 720, 4140, 28337, 85225, 145096, 150325, 94444, 33228, 5040, 21147, 175896, 652703, 1404186, 1908249, 1672524, 921212, 291024

Offset: 1

Alford Arnold, May 07 2006

The initial array of unsigned Stirling numbers of the first kind (filled with an appropriate number of zeros) starts (see A094638)
     1,     0,     0,     0,     0,     0,     0,     0,  ...
     1,     1,     0,     0,     0,     0,     0,     0,  ...
     1,     3,     2,     0,     0,     0,     0,     0,  ...
     1,     6,    11,     6,     0,     0,     0,     0,  ...
     1,    10,    35,    50,    24,     0,     0,     0,  ...
     1,    15,    85,   225,   274,   120,     0,     0,  ...
     1,    21,   175,   735,  1624,  1764,   720,     0,  ...
     1,    28,   322,  1960,  6769, 13132, 13068,  5040,  ...
The Stirling transform is then applied on each individual column. - R. J. Mathar, May 19 2016.

			The array begins
     1;
     2,     1;
     5,     6,     2;
    15,    31,    23,      6;
    52,   160,   195,    110,     24;
   203,   856,  1505,   1365,    634,   120;
   877,  4802, 11312,  14560,  10738,  4284,   720;
  4140, 28337, 85225, 145096, 150325, 94444, 33228, 5040;

Sela Fried, The expected degree of noninvertibility of compositions of functions and a related combinatorial identity, arXiv:2202.13061 [math.CO], 2022. See Corollary 2.6 for a combinatorial identity of a signed version of this sequence.

Cf. A000110 (first column), A000142 (diagonal), A000670 (row sums), A003128 (2nd column), A008275, A008277.

read("transforms"):
A118984 := proc(n,k)
    [seq(0,j=0..k-2), seq( (-1)^k*combinat[stirling1](j+k,j),j=0..n)] ;
    STIRLING(%) ;
    op(n,%) ;
end proc: # R. J. Mathar, May 19 2016

Edited by R. J. Mathar, May 19 2016

A383262 Expansion of e.g.f. f(x)^2 * exp(f(x)) / 2, where f(x) = (exp(3*x) - 1)/3.

Original entry on oeis.org

0, 0, 1, 12, 123, 1270, 13776, 158718, 1944685, 25294338, 348340491, 5064749074, 77528735868, 1246096312188, 20976610875949, 368984700979440, 6767792258171547, 129182459141936566, 2561529454871582772, 52676675861728386114, 1121762199908797394977

Offset: 0

Seiichi Manyama, Apr 21 2025

nonn

Cf. A003128, A383204.

Cf. A286721.

a(n) = sum(k=2, n, 3^(n-k)*binomial(k, 2)*stirling(n, k, 2));

a(n) = Sum_{k=2..n} 3^(n-k) * binomial(k,2) * Stirling2(n,k).

A174436 Partial sums of A039765.

Original entry on oeis.org

0, 0, 4, 35, 275, 2206, 18602, 166191, 1574415, 15788974, 167183914, 1863967135, 21822793215, 267611755414, 3429246890754, 45819357151439, 637076509210383, 9200974953803310, 137799616003035306

Offset: 0

Jonathan Vos Post, Mar 19 2010

nonn

Partial sums of number of edges in the Hasse diagrams for the D-analogs of the partition lattices. The subsequence of primes in this partial sum begins: 45819357151439.

Cf. A003128, A039759, A039765.

a(n) = SUM[i=0..n] A039765(i).

A372624 Expansion of e.g.f. exp(1 - exp(x)) * (exp(x) - 1)^2 / 2.

Original entry on oeis.org

0, 0, 1, 0, -5, -10, 16, 154, 365, -750, -9749, -35222, 20956, 1013220, 6007821, 10272092, -129948837, -1405396426, -6318145964, 7407235766, 371429230721, 3172609248526, 11070816858267, -73488239926510, -1500342260080360, -11917913896465720, -31231507292803479

Offset: 0

Ilya Gutkovskiy, May 07 2024

sign

Cf. A000217, A000225, A000587, A003128, A059606, A081047, A101851.

nmax = 26; CoefficientList[Series[Exp[1 - Exp[x]] (Exp[x] - 1)^2/2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n, k] Binomial[k, 2], {k, 0, n}], {n, 0, 26}]

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * binomial(k,2).

a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,2) * A000587(n-k).

cp's OEIS Frontend

A118984 Triangular T(n,k) which contains in column k >= 0 the elements of the Stirling transform of the unsigned sequence Stirling1(j+k,j), j >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions

A383262 Expansion of e.g.f. f(x)^2 * exp(f(x)) / 2, where f(x) = (exp(3*x) - 1)/3.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A174436 Partial sums of A039765.

Views

Author

Keywords

Comments

Crossrefs

Formula

A372624 Expansion of e.g.f. exp(1 - exp(x)) * (exp(x) - 1)^2 / 2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula