A326324 -id:A326324 - cp's OEIS frontend

A326323 A(n, k) = A_{n}(k) where A_{n}(x) are the Eulerian polynomials, square array read by ascending antidiagonals, for n >= 0 and k >= 0.

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

Offset: 0

Peter Luschny, Jun 27 2019

			Array starts:
  k=0: 1, 1, 1,  1,    1,     1,      1,        1,         1, ... [A000012]
  k=1: 1, 1, 2,  6,   24,   120,    720,     5040,     40320, ... [A000142]
  k=2: 1, 1, 3, 13,   75,   541,   4683,    47293,    545835, ... [A000670]
  k=3: 1, 1, 4, 22,  160,  1456,  15904,   202672,   2951680, ... [A122704]
  k=4: 1, 1, 5, 33,  285,  3081,  40005,   606033,  10491885, ... [A255927]
  k=5: 1, 1, 6, 46,  456,  5656,  84336,  1467376,  29175936, ... [A326324]
  k=6: 1, 1, 7, 61,  679,  9445, 158095,  3088765,  68958295, ... [A384525]
  k=7: 1, 1, 8, 78,  960, 14736, 272448,  5881968, 145105920, ... [A384514]
  k=8: 1, 1, 9, 97, 1305, 21841, 440649, 10386817, 279768825, ...
Seen as a triangle:
  [0], 1
  [1], 1, 1
  [2], 1, 1, 1
  [3], 1, 1, 2,  1
  [4], 1, 1, 3,  6,   1
  [5], 1, 1, 4, 13,  24,    1
  [6], 1, 1, 5, 22,  75,  120,     1
  [7], 1, 1, 6, 33, 160,  541,   720,     1
  [8], 1, 1, 7, 46, 285, 1456,  4683,  5040,     1
  [9], 1, 1, 8, 61, 456, 3081, 15904, 47293, 40320, 1

OEIS Wiki, Eulerian polynomials.

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A384525, A384514.

A := (n, k) -> add(combinat:-eulerian1(k, j)*n^j, j=0..k):
seq(seq(A(n-k, k), k=0..n), n=0..10);
# 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(k!*coeff(ser(n), x, k), k=0..9) od;

a[n_, 0] := 1; a[n_, 1] := n!;
a[n_, k_] := (k - 1)^(n + 1)/k HurwitzLerchPhi[1/k, -n, 0];
(* Alternative: *) a[n_, k_] := Sum[StirlingS2[n, j] (k - 1)^(n - j) j!, {j, 0, n}];
Table[Print[Table[a[n, k], {n, 0, 10}]], {k, 0, 8}]

A(n, k) = Sum_{j=0..k} a(k, j)*n^j where a(k, j) are the Eulerian numbers.
E.g.f.: (n - 1)/(n - exp((n-1)*x)) for n = 0 and n >= 2, 1/(1 - x) if n = 1.
A(n, 0) = 1; A(n, 1) = n!.
A(n, k) = (k - 1)^(n + 1)/k HurwitzLerchPhi(1/k, -n, 0) for k >= 2.
A(n, k) = Sum_{j=0..n} j! * Stirling2(n, j) * (k - 1)^(n - j) for k >= 2.

A355112 Expansion of e.g.f. 4 / (5 - 4x - exp(4x)).

Original entry on oeis.org

1, 2, 12, 112, 1376, 21056, 386688, 8286720, 202958848, 5592199168, 171203895296, 5765504860160, 211811563929600, 8429932686999552, 361312700788375552, 16592261047219388416, 812749365813312487424, 42299637489384965537792, 2330989060564353634271232

Offset: 0

Ilya Gutkovskiy, Jun 19 2022

nonn

Cf. A003576, A006155, A094417, A326324, A343674, A355110, A355111, A355113, A355114.

nmax = 18; CoefficientList[Series[4/(5 - 4 x - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 4^(k-1) * a(n-k).

a(n) ~ n! / ((1 + LambertW(exp(5))) * ((5 - LambertW(exp(5)))/4)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A336952 E.g.f.: 1 / (1 - x * exp(4*x)).

Original entry on oeis.org

1, 1, 10, 102, 1336, 22200, 443664, 10334128, 275060608, 8236914048, 274069953280, 10031110907136, 400520747437056, 17324601073921024, 807023462798608384, 40278407730378332160, 2144307919689898491904, 121291661335680615284736, 7264376142168665821741056

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..375

Column k=4 of A351790.

Cf. A002697, A006153, A235328, A326324, A328183, A336950, A336951.

nmax = 18; CoefficientList[Series[1/(1 - x Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(4 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

seq(n)={ Vec(serlaplace(1 / (1 - x*exp(4*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (4 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 4^(k-1) * a(n-k).
a(n) ~ n! * (4/LambertW(4))^n / (1 + LambertW(4)). - Vaclav Kotesovec, Aug 09 2021

A352071 Expansion of e.g.f. 1 / (1 + log(1 - 4*x) / 4).

Original entry on oeis.org

1, 1, 6, 62, 904, 16984, 390128, 10586736, 331267200, 11738697600, 464539452672, 20302660659456, 971106358760448, 50452643588275200, 2829000818124208128, 170271405502300207104, 10948525752699316371456, 748994717201835804033024, 54315931193865932254543872

Offset: 0

Ilya Gutkovskiy, Mar 02 2022

nonn

Cf. A007840, A047053, A210246, A227917, A326324, A352069.

nmax = 18; CoefficientList[Series[1/(1 + Log[1 - 4 x]/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! (-4)^(n - k), {k, 0, n}], {n, 0, 18}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-4*x)/4))) \\ Michel Marcus, Mar 02 2022

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-4)^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * 4^(k-1) * a(n-k).
a(n) ~ n! * 4^(n+1) * exp(4*n) / (exp(4) - 1)^(n+1). - Vaclav Kotesovec, Mar 03 2022

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

A354751 Expansion of e.g.f. 1 / (1 - log(1 + 4*x) / 4).

Original entry on oeis.org

1, 1, -2, 14, -152, 2264, -42832, 982512, -26484096, 820207488, -28692711168, 1118821622016, -48112717347840, 2261868010650624, -115400220781209600, 6350152838136428544, -374874781697133871104, 23632196147497381625856, -1584445791263626895228928

Offset: 0

Ilya Gutkovskiy, Jun 06 2022

sign

Cf. A006252, A326324, A352071, A354147, A354237, A354264, A354750.

nmax = 18; CoefficientList[Series[1/(1 - Log[1 + 4 x]/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 4^(n - k), {k, 0, n}], {n, 0, 18}]

my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+4*x)/4))) \\ Michel Marcus, Jun 06 2022

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * 4^(n-k).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * (-4)^(k-1) * a(n-k).

A340888 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 4^(n-k-1) * a(k).

Original entry on oeis.org

1, 1, 8, 124, 3456, 150656, 9453056, 807373568, 90066059264, 12716049596416, 2216452086693888, 467465806422867968, 117332539562036035584, 34562989958399757647872, 11807922834511544081973248, 4630865359842075866336067584, 2066370767828213666946077425664

Offset: 0

Ilya Gutkovskiy, Jan 25 2021

nonn

Cf. A102221, A326324, A340886, A340887.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 4^(n - k - 1) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[4/(5 - BesselI[0, 4 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = 4 / (5 - BesselI(0,4*sqrt(x))).

A351810 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 4x)) / (1 - 4x)^2.

Original entry on oeis.org

1, 1, 9, 69, 565, 5305, 56929, 680685, 8902349, 126121313, 1923133433, 31379181461, 544931376229, 10024917092105, 194602995875985, 3972686705253181, 85035210652191485, 1903471938128641457, 44453001710603619369, 1080789854059236415973, 27304602412815047204501

Offset: 0

Ilya Gutkovskiy, Feb 19 2022

nonn

Cf. A004213, A040027, A326324, A351756, A351757, A351811, A351812.

nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x/(1 - 4 x)]/(1 - 4 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] 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k-1) * 4^(k-1) * a(n-k).

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.

A382753 Expansion of e.g.f. 3/(5 - 2exp(3x)).

Original entry on oeis.org

1, 2, 14, 138, 1806, 29562, 580734, 13309578, 348611886, 10272416922, 336326121054, 12112707922218, 475894244100366, 20255443904321082, 928448378212678974, 45597074777924954058, 2388608236671667179246, 132947999835258872046042, 7835059049893316949502494

Offset: 0

Seiichi Manyama, Jun 03 2025

nonn

Wikipedia, Polylogarithm.

Cf. A094417, A326324, A384435.
Cf. A004123, A216794, A384521.
Cf. A201367.

PARI
```
a(n) = (-3)^(n+1)*polylog(-n, 5/2)/5;
```

a(n) = (-3)^(n+1)/5 * Li_{-n}(5/2), where Li_{n}(x) is the polylogarithm function.
a(n) = 3^(n+1)/5 * Sum_{k>=0} k^n * (2/5)^k.
a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * k! * Stirling2(n,k).
a(n) = (2/5) * A201367(n) = (2/5) * Sum_{k=0..n} 5^k * (-3)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 2 * Sum_{k=1..n} 3^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 2 * a(n-1) + 5 * Sum_{k=1..n-1} (-3)^(k-1) * binomial(n-1,k) * a(n-k).

cp's OEIS Frontend

A326323 A(n, k) = A_{n}(k) where A_{n}(x) are the Eulerian polynomials, square array read by ascending antidiagonals, for n >= 0 and k >= 0.

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A355112 Expansion of e.g.f. 4 / (5 - 4*x - exp(4*x)).

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A336952 E.g.f.: 1 / (1 - x * exp(4*x)).

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A352071 Expansion of e.g.f. 1 / (1 + log(1 - 4*x) / 4).

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

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A354751 Expansion of e.g.f. 1 / (1 - log(1 + 4*x) / 4).

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A340888 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 4^(n-k-1) * a(k).

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A351810 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 4*x)) / (1 - 4*x)^2.

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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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A355112 Expansion of e.g.f. 4 / (5 - 4x - exp(4x)).

A351810 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 4x)) / (1 - 4x)^2.

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.

A382753 Expansion of e.g.f. 3/(5 - 2exp(3x)).