A008542 -id:A008542 - cp's OEIS frontend

A347019 E.g.f.: 1 / (1 + 6 * log(1 - x))^(1/6).

1, 1, 8, 114, 2358, 64074, 2157828, 86714592, 4049302404, 215458069428, 12867377875632, 852254389954296, 61998666080311800, 4914000741835488744, 421488717980664846960, 38897664480760253351904, 3843081247426270376211216, 404727487161912602921083536

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

In general, for k >= 1, if e.g.f. = 1 / (1 + k*log(1 - x))^(1/k), then a(n) ~ n! * exp(n/k) / (Gamma(1/k) * k^(1/k) * n^(1 - 1/k) * (exp(1/k) - 1)^(n + 1/k)). - Vaclav Kotesovec, Aug 14 2021

Seiichi Manyama, Table of n, a(n) for n = 0..343

Cf. A007840, A008542, A346978, A346985, A346987, A347015, A347016.

nmax = 17; CoefficientList[Series[1/(1 + 6 Log[1 - x])^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 17}]

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A008542(k).

a(n) ~ n! * exp(n/6) / (Gamma(1/6) * 6^(1/6) * n^(5/6) * (exp(1/6) - 1)^(n + 1/6)). - Vaclav Kotesovec, Aug 14 2021

A347023 E.g.f.: 1 / (1 - 6 * log(1 + x))^(1/6).

Original entry on oeis.org

1, 1, 6, 72, 1254, 28794, 819888, 27869316, 1101032100, 49570797780, 2505156062472, 140417898936336, 8644973807845368, 579908437058338920, 42098286646367326368, 3288252917244250703664, 274974019392668843164176, 24510436934573885695407504, 2319947117871178825560902112

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

In general, for k > 1, if e.g.f. = 1 / (1 - k*log(1 + x))^(1/k), then a(n) ~ n! * exp(1/k^2) / (Gamma(1/k) * k^(1/k) * n^(1 - 1/k) * (exp(1/k) - 1)^(n + 1/k)). - Vaclav Kotesovec, Aug 14 2021

Cf. A006252, A008542, A320343, A346985, A347019, A347020, A347021, A347022.

nmax = 18; CoefficientList[Series[1/(1 - 6 Log[1 + x])^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling1(n,k) * A008542(k).

a(n) ~ n! * exp(1/36) / (Gamma(1/6) * 6^(1/6) * n^(5/6) * (exp(1/6) - 1)^(n + 1/6)). - Vaclav Kotesovec, Aug 14 2021

A153274 Triangle, read by rows, T(n,k) = k^(n+1) * Pochhammer(1/k, n+1).

Original entry on oeis.org

2, 6, 15, 24, 105, 280, 120, 945, 3640, 9945, 720, 10395, 58240, 208845, 576576, 5040, 135135, 1106560, 5221125, 17873856, 49579075, 40320, 2027025, 24344320, 151412625, 643458816, 2131900225, 5925744000, 362880, 34459425, 608608000, 4996616625, 26381811456, 104463111025, 337767408000, 939536222625

Offset: 1

Roger L. Bagula, Dec 22 2008

A Pochhammer function-based triangular sequence.

Row sums are: {2, 21, 409, 14650, 854776, 73920791, 8878927331, 1413788600036, 288152651134776, 73152069870215127, ...}.

			Triangle begins as:
      2;
      6,      15;
     24,     105,      280;
    120,     945,     3640,      9945;
    720,   10395,    58240,    208845,    576576;
   5040,  135135,  1106560,   5221125,  17873856,   49579075;
  40320, 2027025, 24344320, 151412625, 643458816, 2131900225, 5925744000;

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755, A045756.

Cf. A142589, A256268.

Flat(List([1..12], n-> List([1..n], k-> Product([0..n], j-> j*k+1 )))); # G. C. Greubel, Mar 05 2020

[(&*[j*k+1: j in [0..n]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 05 2020

seq(seq( k^(n+1)*pochhammer(1/k, n+1), k=1..n), n=1..12); # G. C. Greubel, Mar 05 2020

Table[Apply[Plus, CoefficientList[j*k^n*Pochhammer[(j+k)/k, n], j]], {n, 12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 05 2020 *)
Table[k^(n+1)*Pochhammer[1/k, n+1], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 05 2020 *)

T(n, k) = prod(j=0, n, j*k+1);
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Mar 05 2020

[[k^(n+1)*rising_factorial(1/k,n+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 05 2020

T(n, k) = k^(n+1) * Pochmammer(1/k, n+1).

T(n, k) = Product_{j=0..n} (j*k + 1). - G. C. Greubel, Mar 05 2020

Edited by G. C. Greubel, Mar 05 2020

A375699 Expansion of e.g.f. 1 / (1 + x^3 * log(1 - x))^(1/6).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 5264, 45360, 409800, 4065600, 77948640, 1183422240, 17527233360, 267109642800, 5422495921920, 110998923235200, 2270809072896000, 47142009514454400, 1116394268619772800, 27963045712157472000, 718066383283082803200

Offset: 0

Seiichi Manyama, Aug 25 2024

nonn

Cf. A351504, A375700, A375701.

Cf. A008542, A351506.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^3*log(1-x))^(1/6)))

a(n) = n!*sum(k=0, n\4, prod(j=0, k-1, 6*j+1)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

a(n) = n! * Sum_{k=0..floor(n/4)} (Product_{j=0..k-1} (6*j+1)) * |Stirling1(n-3*k,k)|/(6^k*(n-3k)!).

A020036 Nearest integer to Gamma(n + 1/6)/Gamma(1/6).

Original entry on oeis.org

1, 0, 0, 0, 1, 6, 29, 177, 1269, 10366, 95020, 966032, 10787363, 131246245, 1728075563, 24481070470, 371296235455, 6002622473192, 103045019123133, 1871984514070245, 35879703186346359, 723574014257984910

Offset: 0

Simon Plouffe

nonn

Gamma(n + 1/6)/Gamma(1/6) = 1, 1/6, 7/36, 91/216, 1729/1296, 43225/7776, 1339975/46656, 49579075/279936, 2131900225/1679616, ... - R. J. Mathar, Sep 04 2016

Cf. A008542, A020081, A020126.

Digits := 64:f := proc(n,x) round(GAMMA(n+x)/GAMMA(x)); end;

With[{c=Gamma[1/6]},Table[Round[Gamma[n+1/6]/c],{n,0,30}]] (* Harvey P. Dale, Jan 08 2022 *)

a(n) = round(gamma(n+1/6)/gamma(1/6)); \\ Michel Marcus, Jun 14 2018

A091546 First column of the array A092077 ((8,2)-Stirling2).

Original entry on oeis.org

1, 56, 10192, 3872960, 2517424000, 2497284608000, 3511182158848000, 6643156644540416000, 16275733779124019200000, 50129260039701979136000000, 189588861470152885092352000000, 863766852858016544480755712000000, 4666068539139005373285042356224000000, 29489553167358513959161467691335680000000

Offset: 1

Wolfdieter Lang, Feb 13 2004

Also seventh column (m=6) of triangle A091543.

Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.

Cf. A007559, A008542, A034689, A091543, A092077.

Cf. A073005, A175379.

a[n_] := 6^(2*n) * Pochhammer[1/6, n] * Pochhammer[1/3, n] / 2; Array[a, 20] (* Amiram Eldar, Aug 30 2025 *)

a(n) = (2^(n-1))*Product_{j=0..n-1} ((3*j+1)*(6*j+1)), n>=1. From eq.12 of the Blasiak et al. reference with r=8, s=2, k=1.
a(n) = (6^(2*n))*risefac(1/6, n)*risefac(1/3, n)/2, n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac6(6*n-5)*fac6(6*n-4)/2, n>=1, with fac6(6*n-5) = A008542(n) and fac6(6*n-4)/2 = A034689(n)= (2^(n-1))*A007559(n), (6-factorials).
a(n) ~ Pi * (6/e)^(2*n) * n^(2*n-1/2) / (Gamma(1/6) * Gamma(1/3)). - Amiram Eldar, Aug 30 2025

A113147 Row 6 of table A113143; equal to INVERT of 6-fold factorials shifted one place right.

Original entry on oeis.org

1, 1, 2, 10, 110, 1954, 47270, 1437562, 52531310, 2239259266, 109021857446, 5966767051354, 362558298692270, 24214789406313442, 1763062297639690790, 138975554045857840570, 11790733617760291994990, 1071215297856049456744642

Offset: 0

Philippe Deléham and Paul D. Hanna, Oct 28 2005

nonn

			A(x) = 1 + x + 2*x^2 + 10*x^3 + 110*x^4 + 1954*x^5 +...
= 1/(1 - x - x^2 - 7*x^3 - 91*x^4 -...- A008542(n)*x^(n+1)
-...).

Cf. A113143, A008542 (6-fold factorials).

{a(n)=local(x=X+X*O(X^n)); A=1/(1-x-x^2*sum(j=0,n,x^j*prod(i=0,j,6*i+1)));return(polcoeff(A,n,X))}

a(n) = Sum_{j=0..k} 6^(k-j)*A111146(k, j).

a(0) = 1; a(n+1) = Sum_{k=0..n} a(k)*A008542(n-k).

A153189 Triangle T(n,k) = Product_{j=0..k} n*j+1.

Original entry on oeis.org

1, 1, 2, 1, 3, 15, 1, 4, 28, 280, 1, 5, 45, 585, 9945, 1, 6, 66, 1056, 22176, 576576, 1, 7, 91, 1729, 43225, 1339975, 49579075, 1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000, 1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625

Offset: 0

Roger L. Bagula, Dec 20 2008

Row sums are: {1, 3, 19, 313, 10581, 599881, 50964103, 6047094369, 954249517513, 193146844030201, 48762935887310811,...}. [Corrected by M. F. Hasler, Oct 28 2014]
This is the lower left triangle of the array A142589. - M. F. Hasler, Oct 28 2014
Row n is a subset of the n-fold factorial sequence for k=0..n. For example, T(8,0..8) is A045755(1..9).  These sequences are listed for n=0..10 in A256268. - Georg Fischer, Feb 15 2020

			Triangle begins as:
  1;
  1, 2;
  1, 3,  15;
  1, 4,  28,  280;
  1, 5,  45,  585,   9945;
  1, 6,  66, 1056,  22176,  576576;
  1, 7,  91, 1729,  43225, 1339975,  49579075;
  1, 8, 120, 2640,  76560, 2756160, 118514880,  5925744000;
  1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625;

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

Cf. A000165, A006882, A007661, A007662, A008544.

Cf. A000142 (row 2), A001147 (3), A007559 (4), A007696 (5), A008548 (6), A008542 (7), A045754 (8), A045755 (9), A045756 (10), A144773 (11), A256268 (combined table).

[(&*[n*j+1: j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 15 2020

seq(seq(mul(n*j+1, j=0..k), k=0..n), n=0..10); # G. C. Greubel, Feb 15 2020

T[n_, k_]= If[n==0 && k==0, 1, Product[n*j+1, {j,0,k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 15 2020 *)
T[n_, k_]:= T[n, k]= If[k<2, 1+k*n, ((1+n*k)*T[n, k-1] + (1+n*k)*(1+n*(k-1))* T[n, k-2])/2]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* Georg Fischer, Feb 17 2020 *)

T(n,k)=prod(j=0,k,n*j+1) \\ M. F. Hasler, Oct 28 2014

[[ product(n*j+1 for j in (0..k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 15 2020

T(n, k) = n^(k+1)*Pochhammer(1/n, k+1).
From Vaclav Kotesovec, Oct 10 2016: (Start)
For fixed n > 0:
T(n, k) ~ sqrt(2*Pi) * n^k * k^(k + 1/2 + 1/n) / (Gamma(1 + 1/n) * exp(k)).
T(n, k) ~ k! * n^k * k^(1/n) / Gamma(1 + 1/n).
(End)
T(n, k) = Sum_{j=0..k+1} (-1)^(k-j+1)*Stirling1(k+1,j)*n^(k-j+1). - G. C. Greubel, Feb 17 2020
T(n, k) = ((1+n*k)*T(n, k-1) + (1+n*k)*(1+n*(k-1))*T(n, k-2))/2. - Georg Fischer, Feb 17 2020

Edited and row 0 added by M. F. Hasler, Oct 28 2014

A131940 Least common multiple of {1, 7, 13, 19, 25, ..., (6n+1)} (A016921).

Original entry on oeis.org

1, 7, 91, 1729, 43225, 1339975, 49579075, 2131900225, 14923301575, 164156317325, 10013535356825, 670906868907275, 48976201430231075, 3869119912988254925, 65775038520800333725, 65775038520800333725, 6380178736517632371325

Offset: 0

Jonathan Vos Post, Oct 05 2007

This is to 6n+1 (A016921) as A051539 is to 4n+1 (A016813). Because 6*9 + 1 = 49 is divisible by 6*1 + 1 = 7, this sequence differs from A008542. a(n) | A008542(n+1).

Cf. A008542, A016813, A016921, A051539.

a = {1}; Do[l = 1; For[j = 1, j < n, j++, l = LCM[l, 6*j + 1]]; AppendTo[a, l], {n, 2, 20}]; a (* Stefan Steinerberger, Oct 07 2007 *)
Join[{1},Table[LCM@@(6*Range[0,n]+1),{n,20}]] (* Harvey P. Dale, Apr 30 2019 *)

More terms from Stefan Steinerberger, Oct 07 2007

A368119 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} (n*j + 1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 15, 24, 1, 1, 1, 5, 28, 105, 120, 1, 1, 1, 6, 45, 280, 945, 720, 1, 1, 1, 7, 66, 585, 3640, 10395, 5040, 1, 1, 1, 8, 91, 1056, 9945, 58240, 135135, 40320, 1, 1, 1, 9, 120, 1729, 22176, 208845, 1106560, 2027025, 362880, 1

Offset: 0

Peter Luschny, Dec 18 2023

A(n, k) is the number of increasing (n + 1)-ary trees on k vertices. (Following a comment of David Callan in A007559.)

			Array A(n, k) starts:
  [0] 1, 1, 1,   1,    1,      1,       1,         1, ...  A000012
  [1] 1, 1, 2,   6,   24,    120,     720,      5040, ...  A000142
  [2] 1, 1, 3,  15,  105,    945,   10395,    135135, ...  A001147
  [3] 1, 1, 4,  28,  280,   3640,   58240,   1106560, ...  A007559
  [4] 1, 1, 5,  45,  585,   9945,  208845,   5221125, ...  A007696
  [5] 1, 1, 6,  66, 1056,  22176,  576576,  17873856, ...  A008548
  [6] 1, 1, 7,  91, 1729,  43225, 1339975,  49579075, ...  A008542
  [7] 1, 1, 8, 120, 2640,  76560, 2756160, 118514880, ...  A045754
  [8] 1, 1, 9, 153, 3825, 126225, 5175225, 253586025, ...  A045755

Index entries for sequences related to factorial numbers.

Rows: A000012, A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755.
Columns: A000012, A000027, A000384, A011199, A011245.
Variant: A256268. Main diagonal: A092985.
Cf. A124320, A008279, A326323.

def A(n, k): return n**k * rising_factorial(1/n, k) if n > 0 else 1
for n in range(9): print([A(n, k) for k in range(8)])

Let rf(n, k) denote the rising factorial and ff(n,k) the falling factorial.
A(n, k) = n^k * rf(1/n, k) if n > 0 else 1.
A(n, k) = (-n)^k * ff(-1/n, k) if n > 0 else 1.
A(n, k) = (n^k * Gamma(k + 1/n)) / Gamma(1/n) for n > 0.
A(n, k) = ((-n)^k * Gamma(1 - 1/n)) / Gamma(1 - 1/n - k) for n > 0.
A(n, k) = k! * [x^k](1 - n*x)^(-1/n).
A(n, k) = [x^k] hypergeom([1, 1/n], [], n*x).
Column n + 1 has a linear recurrence with constant coefficients and signature ((-1)^k*binomial(n+1, n-k) for k=0..n).

cp's OEIS Frontend

A347019 E.g.f.: 1 / (1 + 6 * log(1 - x))^(1/6).

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A347023 E.g.f.: 1 / (1 - 6 * log(1 + x))^(1/6).

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A153274 Triangle, read by rows, T(n,k) = k^(n+1) * Pochhammer(1/k, n+1).

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A375699 Expansion of e.g.f. 1 / (1 + x^3 * log(1 - x))^(1/6).

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A020036 Nearest integer to Gamma(n + 1/6)/Gamma(1/6).

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A091546 First column of the array A092077 ((8,2)-Stirling2).

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A113147 Row 6 of table A113143; equal to INVERT of 6-fold factorials shifted one place right.

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A153189 Triangle T(n,k) = Product_{j=0..k} n*j+1.

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