A046716 -id:A046716 - cp's OEIS frontend

A094869 E.g.f.: exp(5x)/(1-5x)^(1/5).

1, 6, 41, 356, 4401, 78826, 1893481, 56341416, 1978638881, 79749105326, 3622010623401, 182895318578956, 10160561511881041, 615728464210461906, 40414538467581457001, 2855999961062529064976, 216180544920721807887681

Offset: 0

Philippe Deléham, Jun 16 2004

nonn

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n) for x = 1, 2, 3, 4 respectively.

With[{nn=20},CoefficientList[Series[Exp[5x]/(1-5x)^(1/5),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 19 2014 *)

a(n) = Sum_{k = 0..n} A046716(n, k)*5^k.
Conjectured to be D-finite with recurrence: a(n) +(-5*n-1)*a(n-1) +25*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(2*Pi) * 5^n * n^(n - 3/10) / (Gamma(1/5) * exp(n-1)). - Vaclav Kotesovec, Nov 19 2021

A094905 Expansion of e.g.f.: exp(6x)/(1-6x)^(1/6).

Original entry on oeis.org

1, 7, 55, 541, 7585, 157231, 4452247, 157484725, 6594785281, 317357589655, 17222102537911, 1039632137764237, 69073193451776545, 5007661199176196671, 393324947394545293975, 33268708968518818629541

Offset: 0

Philippe Deléham, Jun 16 2004

nonn

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n) for x = 1, 2, 3, 4, 5 respectively.

Cf. A046716.

Cf. A000522, A081367, A094822, A094856, A094869.

With[{nn=20},CoefficientList[Series[Exp[6x]/Surd[1-6x,6],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 15 2022 *)

E.g.f.: exp(6*x)/(1-6*x)^(1/6).
a(n) = Sum_{k = 0..n} A046716(n, k)*6^k.
Conjectured to be D-finite with recurrence: a(n) +(-6*n-1)*a(n-1) +36*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(Pi) * 2^(n + 1/2) * 3^n * n^(n - 1/3) / (Gamma(1/6) * exp(n - 1)). - Vaclav Kotesovec, Nov 19 2021

A059098 Triangle read by rows. T(n, k) = Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..k} (i - j + 1) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 5, 10, 12, 6, 15, 37, 62, 60, 24, 52, 151, 320, 450, 360, 120, 203, 674, 1712, 3120, 3720, 2520, 720, 877, 3263, 9604, 21336, 33600, 34440, 20160, 5040, 4140, 17007, 56674, 147756, 287784, 394800, 352800, 181440, 40320, 21147, 94828

Offset: 0

Vladeta Jovovic, Jan 02 2001

The transpose of this lower unitriangular array is the U factor in the LU decomposition of the Hankel matrix (Bell(i+j-2))A000110(n).%20The%20L%20factor%20is%20A049020%20(see%20Chamberland,%20p.%201672).%20-%20_Peter%20Bala">i,j >= 1, where Bell(n) = A000110(n). The L factor is A049020 (see Chamberland, p. 1672). - _Peter Bala, Oct 15 2023

			Triangle begins:
  [0] [ 1]
  [1] [ 1,    1]
  [2] [ 2,    3,    2]
  [3] [ 5,   10,   12,    6]
  [4] [15,   37,   62,   60,   24]
  [5] [52,  151,  320,  450,  360,  120]
  [6] [203, 674, 1712, 3120, 3720, 2520, 720]
  ...;
E.g.f. for T(n, 2) = (exp(x)-1)^2*(exp(exp(x)-1)) = x^2 + 2*x^3 + 31/12*x^4 + 8/3*x^5 + 107/45*x^6 + 343/180*x^7 + 28337/20160*x^8 + 349/360*x^9 + ...;
E.g.f. for T(n, 3) = (exp(x)-1)^3*(exp(exp(x)-1)) = x^3 + 5/2*x^4 + 15/4*x^5 + 13/3*x^6 + 127/30*x^7 + 1759/480*x^8 + 34961/12096*x^9 + ...

Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.

Cf. A000110(n) = T(n,0), A005493(n) = T(n,1), A059099 (row sums).

Cf. A049020, A001861, A046716.

T := proc(n, k) option remember; `if`(k < 0 or k > n, 0,
      `if`(n = 0, 1, k*T(n-1, k-1) + (k+1)*T(n-1, k) + T(n-1, k+1)))
    end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..15); # Peter Bala, Oct 15 2023

E.g.f. for T(n, k): (exp(x)-1)^k*(exp(exp(x)-1)).
n-th row is M^n*[1,0,0,0,...], where M is a tridiagonal matrix with all 1's in the superdiagonal, (1,2,3,...) in the main and subdiagonals; and the rest zeros. - Gary W. Adamson, Jun 23 2011
T(n, k) = k!*A049020(n, k). - R. J. Mathar, May 17 2016
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j). - Peter Luschny, Dec 06 2023

A094911 Expansion of e.g.f. exp(7x)/(1-7x)^(1/7).

Original entry on oeis.org

1, 8, 71, 778, 12125, 284012, 9241891, 378595022, 18409947641, 1029827400400, 64998958518719, 4565303338264082, 353016345110857429, 29793105387299603252, 2724646021507044539675, 268374407984059193374678

Offset: 0

Philippe Deléham, Jun 17 2004

nonn

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n), A094905(n) for x = 1, 2, 3, 4, 5, 6 respectively.

Cf. A046716, A000522, A081367, A094822, A094856, A094869, A094905.

my(x='x+O('x^20)); Vec(serlaplace(exp(7*x)/(1-7*x)^(1/7))) \\ Michel Marcus, Jan 23 2023

a(n) = Sum_{k = 0..n} A046716(n, k)*7^k.
Conjectured to be D-finite with recurrence: a(n) +(-7*n-1)*a(n-1) +49*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * 7^n / (Gamma(1/7) * exp(n-1) * n^(6/7)). - Vaclav Kotesovec, Nov 19 2021

Corrected by D. S. McNeil, Aug 20 2010

A094935 E.g.f.: exp(8x)/(1-8x)^(1/8).

Original entry on oeis.org

1, 9, 89, 1073, 18321, 476473, 17484457, 813648417, 45054110369, 2872362067433, 206710159889529, 16558892507010961, 1460688620617834801, 140655075719488236057, 14678730623948132120009

Offset: 0

Philippe Deléham, Jun 18 2004

nonn

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n), A094905(n), A094911(n) for x = 1, 2, 3, 4, 5, 6, 7 respectively.

With[{nn=20},CoefficientList[Series[Exp[8x]/Surd[1-8x,8],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 25 2019 *)

a(n) = Sum_{k = 0..n} A046716(n, k)*8^k.
D-finite with recurrence: a(n) +(-8*n-1)*a(n-1) +64*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * 8^n / (Gamma(1/8) * exp(n-1) * n^(7/8)). - Vaclav Kotesovec, Nov 19 2021

A095176 E.g.f.: exp(9x)/(1-9x)^(1/9).

Original entry on oeis.org

1, 10, 109, 1432, 26497, 754894, 30787885, 1603546156, 99602138593, 7128277455538, 576063289419661, 51832424202980320, 5136461847251936929, 555721381650431686582, 65167921144448534609677

Offset: 0

Philippe Deléham, Jun 20 2004

nonn

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n), A094905(n), A094911(n), A094935(n) for x = 1, 2, 3, 4, 5, 6, 7, 8 respectively.
From Vaclav Kotesovec, Nov 19 2021: (Start)
In general, for k > 0, if e.g.f. = exp(k*x) / (1 - k*x)^(1/k), then a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * k^n / (Gamma(1/k) * exp(n-1) * n^(1 - 1/k)).
Equivalently, a(n) ~ n! * exp(1) * k^n / (Gamma(1/k) * n^(1 - 1/k)). (End)

With[{nn=20},CoefficientList[Series[Exp[9x]/Surd[1-9x,9],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 25 2020 *)

a(n) = Sum_{k = 0..n} A046716(n, k)*9^k.
D-finite with recurrence a(n) +(-9*n-1)*a(n-1) +81*(n-1)*a(n-2)=0. - R. J. Mathar, Aug 20 2021
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * 9^n / (Gamma(1/9) * exp(n-1) * n^(8/9)). - Vaclav Kotesovec, Nov 19 2021

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A094869 E.g.f.: exp(5x)/(1-5x)^(1/5).

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A094905 Expansion of e.g.f.: exp(6*x)/(1-6*x)^(1/6).

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A059098 Triangle read by rows. T(n, k) = Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..k} (i - j + 1) for 0 <= k <= n.

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A094911 Expansion of e.g.f. exp(7*x)/(1-7*x)^(1/7).

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A094935 E.g.f.: exp(8x)/(1-8x)^(1/8).

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A095176 E.g.f.: exp(9x)/(1-9x)^(1/9).

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A094905 Expansion of e.g.f.: exp(6x)/(1-6x)^(1/6).

A094911 Expansion of e.g.f. exp(7x)/(1-7x)^(1/7).