A122704 -id:A122704 - cp's OEIS frontend

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

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

A337026 a(n) = (2/3) * Sum_{k>=0} (2*k + 1)^n / 3^k.

Original entry on oeis.org

1, 2, 7, 38, 277, 2522, 27547, 351038, 5112457, 83764082, 1524907087, 30536665238, 667096092637, 15787642820042, 402374890155427, 10987722264846638, 320046586135452817, 9904844539648850402, 324568009210656076567, 11226512280285374623238

Offset: 0

Ilya Gutkovskiy, Aug 11 2020

nonn

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A080253, A122704, A123227.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 2*Exp(x)/(3-Exp(2*x)) ))); // G. C. Greubel, Jun 09 2022

Table[2^(n + 1) HurwitzLerchPhi[1/3, -n, 1/2]/3, {n, 0, 19}]
nmax = 19; CoefficientList[Series[2 Exp[x]/(3 - Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!

def A337026_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 2*exp(x)/(3-exp(2*x)) ).egf_to_ogf().list()
A337026_list(40) # G. C. Greubel, Jun 09 2022

E.g.f.: 2 * exp(x) / (3 - exp(2*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * A122704(k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A123227(k).
a(n) ~ n! * 2^(n+1) / (sqrt(3) * log(3)^(n+1)). - Vaclav Kotesovec, Mar 27 2022
a(n) = 1 + Sum_{k=1..n} 2^(k-1) * binomial(n,k) * a(n-k). - Seiichi Manyama, Dec 24 2023

A354237 Expansion of e.g.f. 1 / (1 - log(1 + 2*x) / 2).

Original entry on oeis.org

1, 1, 0, 2, -8, 64, -592, 6768, -90624, 1395840, -24292608, 471453696, -10094066688, 236340378624, -6007053852672, 164713554069504, -4846361933021184, 152300800682754048, -5091189648734748672, 180386551596145508352, -6752521487083688165376

Offset: 0

Ilya Gutkovskiy, Jun 06 2022

sign

Cf. A006252, A088500, A088501, A122704, A155585, A227917, A354750, A354751.

nmax = 20; CoefficientList[Series[1/(1 - Log[1 + 2 x]/2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 2^(n - k), {k, 0, n}], {n, 0, 20}]

my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+2*x)/2))) \\ Michel Marcus, Jun 06 2022

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * 2^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * (-2)^(k-1) * a(n-k).
a(n) ~ n! * (-1)^(n+1) * 2^(n+1) / (n * log(n)^2) * (1 - (4 + 2*gamma)/log(n) + (12 + 12*gamma + 3*gamma^2 - Pi^2/2)/log(n)^2 + (2*Pi^2*gamma - 32 + 4*Pi^2 - 24*gamma^2 - 8*zeta(3) - 4*gamma^3 - 48*gamma)/log(n)^3 + (80 - 20*Pi^2*gamma + 40*zeta(3)*gamma - 5*Pi^2*gamma^2 + 160*gamma + 5*gamma^4 + 80*zeta(3) + 40*gamma^3 + Pi^4/12 - 20*Pi^2 + 120*gamma^2)/log(n)^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 06 2022

A340886 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 2^(n-k-1) * a(k).

Original entry on oeis.org

1, 1, 6, 76, 1720, 60816, 3096384, 214579296, 19422473088, 2224980891904, 314675568756736, 53849929134122496, 10966912240761425920, 2621246193301011159040, 726608751113679704248320, 231217063994112487051984896, 83713709650818121936828858368

Offset: 0

Ilya Gutkovskiy, Jan 25 2021

nonn

Cf. A102221, A122704, A340887, A340888.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 2^(n - k - 1) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[2/(3 - BesselI[0, 2 Sqrt[2 x]]), {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = 2 / (3 - BesselI(0,2*sqrt(2*x))).

A383149 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = (-1)^k * [m^k] (1/2^(m-n)) * Sum_{k=0..m} k^n * (-1)^m * 3^(m-k) * binomial(m,k).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 12, 9, 1, 0, 66, 75, 18, 1, 0, 480, 690, 255, 30, 1, 0, 4368, 7290, 3555, 645, 45, 1, 0, 47712, 88536, 52290, 12705, 1365, 63, 1, 0, 608016, 1223628, 831684, 249585, 36120, 2562, 84, 1, 0, 8855040, 19019664, 14405580, 5073012, 915705, 87696, 4410, 108, 1

Offset: 0

Seiichi Manyama, Apr 18 2025

			f_0(m) = 1.
f_1(m) =      -m.
f_2(m) =    -3*m +     m^2.
f_3(m) =   -12*m +   9*m^2 -     m^3.
f_4(m) =   -66*m +  75*m^2 -  18*m^3 +    m^4.
f_5(m) =  -480*m + 690*m^2 - 255*m^3 + 30*m^4 - m^5.
Triangle begins:
  1;
  0,     1;
  0,     3,     1;
  0,    12,     9,     1;
  0,    66,    75,    18,     1;
  0,   480,   690,   255,    30,    1;
  0,  4368,  7290,  3555,   645,   45,  1;
  0, 47712, 88536, 52290, 12705, 1365, 63, 1;
  ...

Eric Weisstein's World of Mathematics, Rising Factorial

Columns k=0..3 give A000007, A123227(n-1), A383163, A383164.
Row sums give A122704.
Cf. A001787, A178987, A383150, A383151, A383152, A383155.
Cf. A129062, A209849, A383140.

T(n, k) = sum(j=k, n, 2^(n-j)*stirling(n, j, 2)*abs(stirling(j, k, 1)));

def a_row(n):
    s = sum(2^(n-k)*stirling_number2(n, k)*rising_factorial(x, k) for k in (0..n))
    return expand(s).list()
for n in (0..9): print(a_row(n))

f_n(m) = (1/2^(m-n)) * Sum_{k=0..m} k^n * (-1)^m * 3^(m-k) * binomial(m,k).
T(n,k) = [m^k] f_n(-m).
T(n,k) = Sum_{j=k..n} 2^(n-j) * Stirling2(n,j) * |Stirling1(j,k)|.
T(n,k) = [x^k] Sum_{k=0..n} 2^(n-k) * Stirling2(n,k) * RisingFactorial(x,k).
Sum_{k=0..n} (-1)^k * T(n,k) = f_m(1) = -2^(n-1) for n > 0.
E.g.f. of column k (with leading zeros): g(x)^k / k! with g(x) = -log(1 - (exp(2*x) - 1)/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.

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.

A344913 Table read by rows, T(n, k) (for 0 <= k <= n) = (-2)^(n - k)k!Stirling2(n, k).

Original entry on oeis.org

1, 0, 1, 0, -2, 2, 0, 4, -12, 6, 0, -8, 56, -72, 24, 0, 16, -240, 600, -480, 120, 0, -32, 992, -4320, 6240, -3600, 720, 0, 64, -4032, 28896, -67200, 67200, -30240, 5040, 0, -128, 16256, -185472, 653184, -1008000, 766080, -282240, 40320

Offset: 0

Peter Luschny, Aug 14 2021

			Table starts:
[0] 1;
[1] 0,    1;
[2] 0,   -2,     2;
[3] 0,    4,   -12,       6;
[4] 0,   -8,    56,     -72,     24;
[5] 0,   16,  -240,     600,   -480,      120;
[6] 0,  -32,   992,   -4320,   6240,    -3600,    720;
[7] 0,   64, -4032,   28896, -67200,    67200, -30240,    5040;
[8] 0, -128, 16256, -185472, 653184, -1008000, 766080, -282240, 40320.

Cf. A155585 (row sums), A122704 (alternating row sums, signed), A278075 (signed Fubini polynomials), A000142 (main diagonal), A048993 (Stirling2).

T := (n, k) -> (-2)^(n - k)*k!*Stirling2(n, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

T(n, k) = (-2)^(n - k)*k!*stirling(n, k, 2); \\ Michel Marcus, Aug 14 2021

T(n, k) = 2^(n - k)*Sum_{j=0..n} (-1)^(n - j)*binomial(k, j)*j^n.

Let row(n, x) be the n-th row polynomial, then B(n) = row(n-1, 1)*n / (4^n - 2^n) is the n-th Bernoulli number (with B(1) = 1/2) for n >= 1.

A371298 E.g.f. satisfies A(x) = 2/(3 - exp(2xA(x)^2)).

Original entry on oeis.org

1, 1, 8, 124, 2928, 93496, 3773536, 184354752, 10580324096, 697840047616, 52018550966784, 4324989984168448, 396842631019350016, 39833949803142014976, 4342129457277000261632, 510808184298890239393792, 64504327889586673547673600, 8703038855093947990994452480

Offset: 0

Seiichi Manyama, Mar 18 2024

nonn

Cf. A122704, A370894.

Cf. A371296.

a(n) = sum(k=0, n, 2^(n-k)*(2*n+k)!*stirling(n, k, 2))/(2*n+1)!;

a(n) = (1/(2*n+1)!) * Sum_{k=0..n} 2^(n-k) * (2*n+k)! * Stirling2(n,k).

cp's OEIS Frontend

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

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A337026 a(n) = (2/3) * Sum_{k>=0} (2*k + 1)^n / 3^k.

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A354237 Expansion of e.g.f. 1 / (1 - log(1 + 2*x) / 2).

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A340886 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 2^(n-k-1) * a(k).

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A383149 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = (-1)^k * [m^k] (1/2^(m-n)) * Sum_{k=0..m} k^n * (-1)^m * 3^(m-k) * binomial(m,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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A344913 Table read by rows, T(n, k) (for 0 <= k <= n) = (-2)^(n - k)*k!*Stirling2(n, k).

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A371298 E.g.f. satisfies A(x) = 2/(3 - exp(2*x*A(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.

A344913 Table read by rows, T(n, k) (for 0 <= k <= n) = (-2)^(n - k)k!Stirling2(n, k).

A371298 E.g.f. satisfies A(x) = 2/(3 - exp(2xA(x)^2)).