A326323 -id:A326323 - cp's OEIS frontend

A326324 a(n) = A_{5}(n) where A_{m}(x) are the Eulerian polynomials as defined in A326323.

1, 1, 6, 46, 456, 5656, 84336, 1467376, 29175936, 652606336, 16219458816, 443419545856, 13224580002816, 427278468668416, 14867050125981696, 554245056343668736, 22039796215883268096, 931198483176870608896, 41658202699736550014976, 1967160945260218035798016

Offset: 0

Peter Luschny, Jun 27 2019

nonn

See A326323 for the more general formulas.

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326323.

seq(add(combinat:-eulerian1(n,k)*5^k, k=0..n), n=0..20);
# Alternative:
egf := 4/(5 - exp(4*x)): ser := series(egf, x, 21):
seq(k!*coeff(ser, x, k), k=0..20);

a[1] := 1; a[n_] := 4^(n + 1)/5 HurwitzLerchPhi[1/5, -n, 0];
Table[a[n], {n, 0, 20}]
(* Alternative: *)
s[n_] := Sum[StirlingS2[n, j] 4^(n - j) j!, {j, 0, n}];
Table[s[n], {n, 0, 20}]

a(n) ~ n!/5 * (4/log(5))^(n+1). - Vaclav Kotesovec, Aug 09 2021

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * 4^(k-1) * a(n-k). - Ilya Gutkovskiy, Feb 04 2022

Corrected after notice from Jean-François Alcover by Peter Luschny, Jul 13 2019

A255927 a(n) = (3/4) * Sum_{k>=0} (3*k)^n/4^k.

Original entry on oeis.org

1, 1, 5, 33, 285, 3081, 40005, 606033, 10491885, 204343641, 4422082005, 105265315233, 2733583519485, 76902684021801, 2329889536156005, 75629701786875633, 2618654297178083085, 96336948993312237561, 3752590641305604502005, 154294551397830418471233, 6677999524135208461382685

Offset: 0

Karol A. Penson, Sep 03 2015

nonn

			a(5) = 729*hypergeom([2,2,2,2,2],[1,1,1,1],1/4)/16 = 3081.

Alois P. Heinz, Table of n, a(n) for n = 0..400
P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the Log-normal distribution, arXiv:quant-ph/0303030, 2003.
P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the Log-normal distribution, J. Phys. A: Math. Gen. 36, (2003), L273.
Eric Weisstein's World of Mathematics, Lerch Transcendent

Cf. A000670, A094418, A004123 A032033, A094417, A122704, A326323.

S:= series(3/(4-exp(3*x)), x, 51):
seq(coeff(S,x,n)*n!, n=0..50); # Robert Israel, Sep 03 2015
seq(add(combinat:-eulerian1(n,k)*4^k, k=0..n), n=0..20); # Peter Luschny, Jun 27 2019

a[n_] := 3^(n+1)/4 HurwitzLerchPhi[1/4, -n, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Sep 18 2018 *)
Eulerian1[0, 0] = 1; Eulerian1[n_, k_] := Sum[(-1)^j (k-j+1)^n Binomial[n+1, j], {j, 0, k+1}]; Table[Sum[Eulerian1[n, k] 4^k, {k, 0, n}], {n, 0, 20}] (* Jean-François Alcover, Jul 13 2019, after Peter Luschny *)

a(n) = sum(k=0, n, stirling(n,k,2)*k!*3^(n-k)); \\ Michel Marcus, Sep 03 2015

a(n) = Sum_{k>=0} Stirling2(n,k)*k!*3^(n-k).
E.g.f.: 3/(4-exp(3*x)).
Special values of the generalized hypergeometric function n_F_(n-1):
a(n) = (3^(n+1)/16) * hypergeom([2,2,..2],[1,1,..1],1/4), where the sequence in the first square bracket ("upper" parameters) has n elements all equal to 2 whereas the sequence in the second square bracket ("lower" parameters) has n-1 elements all equal to 1.
Example: a(5) = 729 * hypergeom([2,2,2,2,2],[1,1,1,1],1/4)/16 = 3081.
a(n) is the n-th moment of the discrete weight function W(x) = (3/4)*sum(k>=0, Dirac(x-3*k)/4^k), n>=1.
a(n) ~ n! * 3^(n+1) / ((log(2))^(n+1) * 2^(n+3)). - Vaclav Kotesovec, Jul 09 2018
G.f.: Sum_{j>=0} j!*x^j / Product_{k=1..j} (1 - 3*k*x). - Ilya Gutkovskiy, Apr 04 2019
a(n) = A_{4}(n) where A_{n}(x) are the Eulerian polynomials as defined in A326323. - Peter Luschny, Jun 27 2019

a(0)=1 prepended by Alois P. Heinz, Sep 18 2018

A384514 Expansion of e.g.f. 6/(7 - exp(6*x)).

Original entry on oeis.org

1, 1, 8, 78, 960, 14736, 272448, 5881968, 145105920, 4026744576, 124159039488, 4211132779008, 155814875873280, 6245695887446016, 269610827961212928, 12469729905669224448, 615184657168540631040, 32246522356406129197056, 1789714914567248392224768

Offset: 0

Seiichi Manyama, Jun 01 2025

nonn

Wikipedia, Polylogarithm.

Cf. A326323.

Cf. A094419, A384521, A384522, A384523, A384524.

PARI
```
a(n) = (-6)^(n+1)*polylog(-n, 7)/7;
```

a(n) = (-6)^(n+1)/7 * Li_{-n}(7), where Li_{n}(x) is the polylogarithm function.
a(n) = 6^(n+1) * Sum_{k>=0} k^n * (1/7)^(k+1).
a(n) = Sum_{k=0..n} 6^(n-k) * k! * Stirling2(n,k).
a(n) = (1/7) * Sum_{k=0..n} 7^k * (-6)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = Sum_{k=1..n} 6^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) + 7 * Sum_{k=1..n-1} (-6)^(k-1) * binomial(n-1,k) * a(n-k).

A331690 a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * n^(n - k).

Original entry on oeis.org

1, 1, 4, 33, 456, 9445, 272448, 10386817, 503758720, 30202999821, 2189000524800, 188349613075393, 18954958449853440, 2203304642871358741, 292675996808408743936, 44022321302156791898625, 7438113993194856900034560, 1401876939543892434209075581

Offset: 0

Ilya Gutkovskiy, Jan 24 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..271

Cf. A000670, A063170, A086914, A094420, A122704, A122778, A229234, A255927, A301419, A326323, A326324.

Join[{1}, Table[Sum[StirlingS2[n, k] k! n^(n - k), {k, 0, n}], {n, 1, 17}]]
Table[SeriesCoefficient[Sum[k! x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[n^(n + 1) PolyLog[-n, 1/(n + 1)]/(n + 1), {n, 1, 17}]]

a(n) = sum(k=0, n, stirling(n, k, 2)*k!*n^(n-k)); \\ Michel Marcus, Jan 24 2020

a(n) = [x^n] Sum_{k>=0} k! * x^k / Product_{j=1..k} (1 - n*j*x).
a(n) = n! * [x^n] n / (1 + n - exp(n*x)) for n > 0.
a(n) = n^(n + 1) * Sum_{k>=1} k^n / (n + 1)^(k + 1) for n > 0.
a(n) ~ n! * n^(n+1) / ((n+1) * log(n+1)^(n+1)). - Vaclav Kotesovec, Jun 06 2022

A332700 A(n, k) = Sum_{j=0..n} j!Stirling2(n, j)(k-1)^(n-j), for n >= 0 and k >= 0, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 13, 4, 1, 1, 1, 120, 75, 22, 5, 1, 1, 1, 720, 541, 160, 33, 6, 1, 1, 1, 5040, 4683, 1456, 285, 46, 7, 1, 1, 1, 40320, 47293, 15904, 3081, 456, 61, 8, 1, 1, 1, 362880, 545835, 202672, 40005, 5656, 679, 78, 9, 1, 1

Offset: 0

Peter Luschny, Feb 28 2020

			Array begins:
[0] 1, 1,       1,       1,        1,         1,         1, ...    A000012
[1] 1, 1,       1,       1,        1,         1,         1, ...    A000012
[2] 1, 2,       3,       4,        5,         6,         7, ...    A000027
[3] 1, 6,       13,      22,       33,        46,        61, ...   A028872
[4] 1, 24,      75,      160,      285,       456,       679, ...
[5] 1, 120,     541,     1456,     3081,      5656,      9445, ...
[6] 1, 720,     4683,    15904,    40005,     84336,     158095, ...
[7] 1, 5040,    47293,   202672,   606033,    1467376,   3088765, ...
[8] 1, 40320,   545835,  2951680,  10491885,  29175936,  68958295, ...
[9] 1, 362880,  7087261, 48361216, 204343641, 652606336, 1731875605, ...
       A000142, A000670, A122704,  A255927,   A326324, ...
Seen as a triangle:
[0] [1]
[1] [1, 1]
[2] [1, 1,     1]
[3] [1, 2,     1,     1]
[4] [1, 6,     3,     1,     1]
[5] [1, 24,    13,    4,     1,    1]
[6] [1, 120,   75,    22,    5,    1,   1]
[7] [1, 720,   541,   160,   33,   6,   1,  1]
[8] [1, 5040,  4683,  1456,  285,  46,  7,  1, 1]
[9] [1, 40320, 47293, 15904, 3081, 456, 61, 8, 1, 1]

Paolo Xausa, Table of n, a(n) for n = 0..11475 (antidiagonals 0..150 of the square, flattened).

The matrix transpose of A326323.

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A000027, A028872.

# Prints array by row.
A := (n, k) -> add(combinat:-eulerian1(n, j)*k^j, j=0..n):
seq(print(seq(A(n,k), k=0..10)), n=0..8);
# Alternative:
egf := n -> `if`(n=1, 1/(1-x), (n-1)/(n - exp((n-1)*x))):
ser := n -> series(egf(n), x, 21):
for n from 0 to 6 do seq(n!*coeff(ser(k), x, n), k=0..9) od;
# Or:
A := (n, k) -> if k = 0 or n = 0 then 1 elif k = 1 then n! else
polylog(-n, 1/k)*(k-1)^(n+1)/k fi:
for n from 0 to 6 do seq(A(n, k), k=0..9) od;

A332700[n_, k_] := n! + Sum[j! StirlingS2[n, j] (k-1)^(n-j), {j, n-1}];
Table[A332700[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 01 2024 *)

def T(n, k):
    return sum(factorial(j)*stirling_number2(n, j)*(k-1)^(n-j) for j in range(n+1))
for n in range(8): print([T(n, k) for k in range(8)])

A(n, k) = Sum_{j=0..n} E(n, j)*k^j, where E(n, k) = A173018(n, k).
A(n, 1) = n!*[x^n] 1/(1-x).
A(n, k) = n!*[x^n] (k-1)/(k - exp((k-1)*x)) for k != 1.
A(n, k) = PolyLog(-n, 1/k)*(k-1)^(n+1)/k for k >= 2.

A384525 Expansion of e.g.f. 5/(6 - exp(5*x)).

Original entry on oeis.org

1, 1, 7, 61, 679, 9445, 158095, 3088765, 68958295, 1731875605, 48328686175, 1483501074925, 49677478279975, 1802159471217925, 70406303657894575, 2947087948180076125, 131584088098220272375, 6242270620707298139125, 313548981075158413477375

Offset: 0

Seiichi Manyama, Jun 01 2025

nonn

Wikipedia, Polylogarithm.

Cf. A326323.

PARI
```
a(n) = (-5)^(n+1)*polylog(-n, 6)/6;
```

a(n) = (-5)^(n+1)/6 * Li_{-n}(6), where Li_{n}(x) is the polylogarithm function.
a(n) = 5^(n+1) * Sum_{k>=0} k^n * (1/6)^(k+1).
a(n) = Sum_{k=0..n} 5^(n-k) * k! * Stirling2(n,k).
a(n) = (1/6) * Sum_{k=0..n} 6^k * (-5)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = Sum_{k=1..n} 5^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) + 6 * Sum_{k=1..n-1} (-5)^(k-1) * binomial(n-1,k) * a(n-k).

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

A326324 a(n) = A_{5}(n) where A_{m}(x) are the Eulerian polynomials as defined in A326323.

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A255927 a(n) = (3/4) * Sum_{k>=0} (3*k)^n/4^k.

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A384514 Expansion of e.g.f. 6/(7 - exp(6*x)).

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A331690 a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * n^(n - k).

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A332700 A(n, k) = Sum_{j=0..n} j!*Stirling2(n, j)*(k-1)^(n-j), for n >= 0 and k >= 0, read by ascending antidiagonals.

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A384525 Expansion of e.g.f. 5/(6 - exp(5*x)).

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

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A332700 A(n, k) = Sum_{j=0..n} j!Stirling2(n, j)(k-1)^(n-j), for n >= 0 and k >= 0, read by ascending antidiagonals.