A005012 -id:A005012 - cp's OEIS frontend

A241578 Square array read by antidiagonals upwards: T(n,k) = Sum_{j=1..k} n^(k-j)*Stirling_2(k,j) (n >= 0, k >= 1).

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 15, 1, 1, 5, 19, 49, 52, 1, 1, 6, 29, 109, 257, 203, 1, 1, 7, 41, 201, 742, 1539, 877, 1, 1, 8, 55, 331, 1657, 5815, 10299, 4140, 1, 1, 9, 71, 505, 3176, 15821, 51193, 75905, 21147, 1, 1, 10, 89, 729, 5497, 35451, 170389, 498118, 609441, 115975, 1

Offset: 0

N. J. A. Sloane, Apr 29 2014

			Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, ...
1, 3, 11, 49, 257, 1539, 10299, 75905, 609441, 5284451, 49134923, 487026929, ...
1, 4, 19, 109, 742, 5815, 51193, 498118, 5296321, 60987817, 754940848, 9983845261, ...
1, 5, 29, 201, 1657, 15821, 170389, 2032785, 26546673, 376085653, 5736591885, 93614616409, ...
1, 6, 41, 331, 3176, 35451, 447981, 6282416, 96546231, 1611270851, 28985293526, 558413253581, ...
1, 7, 55, 505, 5497, 69823, 1007407, 16157905, 284214097, 5432922775, 112034017735, 2476196276617, ...
1, 8, 71, 729, 8842, 125399, 2026249, 36458010, 719866701, 15453821461, 358100141148, 8899677678109, ...
...

Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]

Three versions of this array are A111673, A241578, A241579.

Rows and columns give A000110, A004211, A004212, A004213, A005011, A005012, A028387, A241577.

with(combinat):
T:=(n,k)->add(n^(k-j)*stirling2(k,j),j=1..k);
r:=n->[seq(T(n,k),k=1..12)];
for n from 0 to 8 do lprint(r(n)); od:

A337042 a(n) = exp(-1/6) * Sum_{k>=0} (6k - 1)^n / (6^k k!).

Original entry on oeis.org

1, 0, 6, 36, 324, 3456, 43416, 618192, 9778320, 169827840, 3210376032, 65540155968, 1435094563392, 33510354739200, 830486180748672, 21756166766173440, 600339119317643520, 17394883290643709952, 527782830161632077312, 16727350847049194775552

Offset: 0

Ilya Gutkovskiy, Aug 12 2020

nonn

In general, if m >= 1, b <> 0 and e.g.f. = exp(m*exp(b*x) + r*x + s) then a(n) ~ b^n * n^(n + r/b) * exp(n/LambertW(n/m) - n + s) / (m^(r/b) * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n + r/b)). - Vaclav Kotesovec, Jun 28 2022

Cf. A000296, A003578, A005012, A337038, A337039, A337040, A337041, A337043.

nmax = 19; CoefficientList[Series[Exp[(Exp[6 x] - 1)/6 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 6^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 19}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 6^k BellB[k, 1/6], {k, 0, n}], {n, 0, 19}]

G.f. A(x) satisfies: A(x) = (1 - 6*x + x*A(x/(1 - 6*x))) / (1 - 5*x - 6*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 6*j*x/(1 + x)).
E.g.f.: exp((exp(6*x) - 1) / 6 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 6^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A005012(k).
a(n) ~ 6^(n - 1/6) * n^(n - 1/6) * exp(n/LambertW(6*n) - n - 1/6) / (sqrt(1 + LambertW(6*n)) * LambertW(6*n)^(n - 1/6)). - Vaclav Kotesovec, Jun 26 2022

A351152 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 6x)) / (1 - 6x).

Original entry on oeis.org

1, 0, 1, 6, 37, 240, 1693, 13446, 122329, 1261104, 14332681, 175123446, 2267871517, 30981705984, 446571784261, 6798161166486, 109220619908593, 1846729159654560, 32726973173941585, 605358657750562470, 11648701234354836565, 232655173657593759312

Offset: 0

Ilya Gutkovskiy, Feb 02 2022

nonn

Shifts 2 places left under 6th-order binomial transform.

Cf. A000994, A005012, A351057, A351143, A351144, A351150, A351151.

nmax = 21; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 6 x)]/(1 - 6 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 6^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 21}]

a(0) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 6^k * a(n-k-2).

A132164 Row sums of triangle A134141 (S1p(7)).

Original entry on oeis.org

1, 1, 8, 78, 918, 12846, 209616, 3909228, 81859548, 1897344828, 48135826656, 1325008302696, 39292978029768, 1247949491330088, 42236558731574208, 1516738194700667856, 57573649342673292816, 2302425590703685075728, 96720470167595138898048

Offset: 0

Wolfdieter Lang, Oct 12 2007

Alois P. Heinz, Table of n, a(n) for n = 0..409

Cf. A132165 (alternating row sum of A134141), A049428.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+5)!/6!*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017

a[n_]:=a[n]=If[n==0, 1, Sum[Binomial[n - 1, j - 1] (j + 5)!/6! a[n - j], {j, n}]]; Table[a[n], {n, 0, 25}] (* Indranil Ghosh, Aug 02 2017, after Maple code *)

a(n)= sum(A134141(n,m),m=1..n),n>=1.
E.g.f.: exp((1-(1-x)^6)/(6*(1-x)^6)). Cf. e.g.f. first column of A134141.
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A005012(k).
a(n) = (1/exp(1/6)) * (-1)^n * n! * Sum_{k>=0} binomial(-6*k,n)/(6^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A241579 Square array read by antidiagonals downwards: T(n,k) = Sum_{j=1..k} n^(k-j)*Stirling_2(k,j) (n >= 0, k >= 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 15, 11, 4, 1, 1, 52, 49, 19, 5, 1, 1, 203, 257, 109, 29, 6, 1, 1, 877, 1539, 742, 201, 41, 7, 1, 1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1, 21147, 75905, 51193, 15821, 3176, 505, 71, 9, 1, 1, 115975, 609441, 498118, 170389, 35451, 5497, 729, 89, 10, 1

Offset: 0

N. J. A. Sloane, Apr 29 2014

			Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, ...
1, 3, 11, 49, 257, 1539, 10299, 75905, 609441, 5284451, 49134923, 487026929, ...
1, 4, 19, 109, 742, 5815, 51193, 498118, 5296321, 60987817, 754940848, 9983845261, ...
1, 5, 29, 201, 1657, 15821, 170389, 2032785, 26546673, 376085653, 5736591885, 93614616409, ...
1, 6, 41, 331, 3176, 35451, 447981, 6282416, 96546231, 1611270851, 28985293526, 558413253581, ...
1, 7, 55, 505, 5497, 69823, 1007407, 16157905, 284214097, 5432922775, 112034017735, 2476196276617, ...
1, 8, 71, 729, 8842, 125399, 2026249, 36458010, 719866701, 15453821461, 358100141148, 8899677678109, ...
...

A. Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
A. Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]

Three versions of this array are A111673, A241578, A241579.

Rows and columns give A000110, A004211, A004212, A004213, A005011, A005012, A028387, A241577.

with(combinat):
T:=(n,k)->add(n^(k-j)*stirling2(k,j),j=1..k);
r:=n->[seq(T(n,k),k=1..12)];
for n from 0 to 8 do lprint(r(n)); od:

A351057 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 6x)) / (1 - 6x).

Original entry on oeis.org

1, 1, 1, 7, 49, 349, 2593, 20755, 184609, 1851289, 20735041, 253471039, 3310505425, 45630322741, 660993079393, 10065000586507, 161262522401089, 2717539655666353, 48053169836707969, 888408313419305719, 17108882037936283249, 342144175940842590349, 7089944927940141776545

Offset: 0

Ilya Gutkovskiy, Jan 30 2022

nonn

Shifts 2 places left under 6th-order binomial transform.

Cf. A005012, A007472, A007476, A351049, A351050, A351056.

nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - 6 x)]/(1 - 6 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 6^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 6^k * a(n-k-2).

A351161 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 6x)) / (1 - 6x).

Original entry on oeis.org

0, 1, 0, 1, 12, 109, 900, 7309, 62280, 590185, 6402360, 78347593, 1042633908, 14648616757, 214421295132, 3266839420021, 52041902492496, 870810496011793, 15326196662766384, 283049655668743249, 5460180803581446684, 109489002283248831037, 2273856664328893182324

Offset: 0

Ilya Gutkovskiy, Feb 03 2022

nonn

Shifts 2 places left under 6th-order binomial transform.

Cf. A000995, A005012, A351028, A351053, A351057, A351128, A351132, A351152.

nmax = 22; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 6 x)]/(1 - 6 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 6^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

a(0) = 0, a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 6^k * a(n-k-2).

A111673 Triangle, generated from A111579.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 15, 11, 4, 1, 1, 1, 52, 49, 19, 5, 1, 1, 1, 203, 257, 109, 29, 6, 1, 1, 1, 877, 1539, 742, 201, 41, 7, 1, 1, 1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1, 1, 21147, 75905, 51193, 15821, 3176, 505, 71, 9, 1, 1, 1, 115975, 609441, 498118, 170389, 35451, 5497, 729, 89, 10, 1, 1

Offset: 0

Gary W. Adamson, Aug 14 2005

Columns are inverse binomial transforms of columns (k>0) of A111579.

			First few rows of the triangle are:
  1,
  1, 1,
  1, 1, 1,
  1, 2, 1, 1,
  1, 5, 3, 1, 1,
  1, 15, 11, 4, 1, 1,
  1, 52, 49, 19, 5, 1, 1,
  1, 203, 257, 109, 29, 6, 1, 1,
  1, 877, 1539, 742, 201, 41, 7, 1, 1,
  1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1,
  ...
Inverse binomial transform of column 2 of A111579 (1, 2, 5, 15, 52, 203...) = column 2 (1, 1, 2, 5, 15, 52...).

A. Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
A. Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]

Cf. A111579, A008277.
For two other versions of this triangle see A241578, A241579.
Rows and columns give A000110, A004211, A004212, A004213, A005011, A005012, A028387, A241577.

More terms from N. J. A. Sloane, Apr 29 2014

A351812 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 6x)) / (1 - 6x)^2.

Original entry on oeis.org

1, 1, 13, 139, 1531, 19021, 271453, 4358179, 76896931, 1471496341, 30333401893, 670125430219, 15784342627531, 394467249489661, 10415430504486733, 289527454704656659, 8447556960083354131, 258008113711846390981, 8228947382557338981973, 273472796359924298018299

Offset: 0

Ilya Gutkovskiy, Feb 19 2022

nonn

Cf. A005012, A040027, A351756, A351757, A351810, A351811.

nmax = 19; A[] = 0; Do[A[x] = 1 + x A[x/(1 - 6 x)]/(1 - 6 x)^2 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k - 1] 6^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k-1) * 6^(k-1) * a(n-k).

cp's OEIS Frontend

A241578 Square array read by antidiagonals upwards: T(n,k) = Sum_{j=1..k} n^(k-j)*Stirling_2(k,j) (n >= 0, k >= 1).

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A337042 a(n) = exp(-1/6) * Sum_{k>=0} (6*k - 1)^n / (6^k * k!).

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A351152 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 6*x)) / (1 - 6*x).

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A132164 Row sums of triangle A134141 (S1p(7)).

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A241579 Square array read by antidiagonals downwards: T(n,k) = Sum_{j=1..k} n^(k-j)*Stirling_2(k,j) (n >= 0, k >= 1).

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A351057 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 6*x)) / (1 - 6*x).

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A351161 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 6*x)) / (1 - 6*x).

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A111673 Triangle, generated from A111579.

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A351812 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 6*x)) / (1 - 6*x)^2.

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A337042 a(n) = exp(-1/6) * Sum_{k>=0} (6k - 1)^n / (6^k k!).

A351152 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 6x)) / (1 - 6x).

A351057 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 6x)) / (1 - 6x).

A351161 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 6x)) / (1 - 6x).

A351812 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 6x)) / (1 - 6x)^2.