A298851 -id:A298851 - cp's OEIS frontend

A350366 a(n) = [x^n] Product_{k=1..n} (1 + kx)/(1 - kx).

1, 2, 18, 312, 8000, 271770, 11502162, 583036832, 34437042432, 2322677883330, 176137593178250, 14835018315726312, 1373972097646792800, 138787120025382437882, 15184417945878202716450, 1788809909368939651651200, 225755544056485027686459392

Offset: 0

Seiichi Manyama, Dec 27 2021

nonn

Cf. A007820, A008275, A008277, A298851, A384043, A384044.

a[n_] := Coefficient[Series[Product[(1 + k*x)/(1 - k*x), {k, 1, n}], {x, 0, n}], x, n]; Array[a, 17, 0] (* Amiram Eldar, Dec 27 2021 *)
Table[Sum[(-1)^(n - k)*StirlingS1[n + 1, k + 1] * StirlingS2[k + n, n], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 29 2021 *)

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

a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n+1, k+1) * Stirling2(k+n, n).

a(n) ~ c * d^n * (n-1)!, where d = (1+r) / ((-1 + exp(r + LambertW(-1, -exp(-r)*r))) * LambertW(-exp(-1-r)*(1+r))) = 8.406107401279769476199925123910168..., r = 0.7545302104650497245839827141610818561001159135034... is the root of the equation r*(1 + r + LambertW(-exp(-1 - r)*(1 + r))) = -(1 + r)*(r + LambertW(-1, -exp(-r)*r)) and c = 0.281498742412700978029375818376931142913157133987685... - Vaclav Kotesovec, Dec 29 2021

A350376 a(n) = [x^n] Product_{k=1..n} 1/(1 - k*x)^2.

Original entry on oeis.org

1, 2, 23, 480, 14627, 587580, 29331038, 1750923328, 121673580435, 9648709656300, 859874920598850, 85078769750118144, 9254316901029412110, 1097635452798476278232, 140986468651523106196060, 19496446561112852736019200, 2887977880849714395963280515

Offset: 0

Seiichi Manyama, Dec 27 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A007820, A008277, A129256, A298851, A350366, A383862, A384060.

a[n_] := Coefficient[Series[Product[1/(1 - k*x)^2, {k, 1, n}], {x, 0, n}], x, n]; Array[a, 17, 0] (* Amiram Eldar, Dec 28 2021 *)
Table[Sum[StirlingS2[n + k, n]*StirlingS2[2*n - k, n], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 29 2021 *)

a(n) = sum(k=0, n, stirling(n+k, n, 2)*stirling(2*n-k, n, 2));

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

a(n) ~ c * d^n * (n-1)!, where d = 27 / (4*LambertW(-3*exp(-3/2)/2)^2 * (3 + 2*LambertW(-3*exp(-3/2)/2))) = 9.858422414446789720857925020919293523149... and c = sqrt(3/(-LambertW(-3*exp(-3/2)/2) * (1 + LambertW(-3*exp(-3/2)/2)))) / (4*Pi) = 0.28482428628793763109169664913715827647091747... - Vaclav Kotesovec, Dec 28 2021, updated May 14 2025

A269945 Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 21, 14, 1, 0, 1, 85, 147, 30, 1, 0, 1, 341, 1408, 627, 55, 1, 0, 1, 1365, 13013, 11440, 2002, 91, 1, 0, 1, 5461, 118482, 196053, 61490, 5278, 140, 1, 0, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1

Offset: 0

Peter Luschny, Mar 22 2016

Also known as central factorial numbers T(2*n, 2*k) (cf. A036969).

The analog for the Stirling cycle numbers is A269944.

			Triangle starts:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1,   1]
  [3] [0, 1,   5,    1]
  [4] [0, 1,  21,   14,   1]
  [5] [0, 1,  85,  147,  30,  1]
  [6] [0, 1, 341, 1408, 627, 55, 1]

M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
Peter Luschny, The P-transform.
Peter Luschny, The Partition Transform -- A SageMath Jupyter Notebook.

Columns k=0..5 give A000007, A000012, A002450(n-1), A002451(n-3), A383838(n-4), A383840(n-5).
Variants are: A008957, A036969.
Cf. A007318 (order 0), A048993 (order 1), A269948 (order 3).
Cf. A000330 (subdiagonal), A002450 (column 2), A135920 (row sums), A269941, A269944 (Stirling cycle), A298851 (central terms).

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + k^2*T(n-1, k))) end:
for n from 0 to 9 do seq(T(n, k), k=0..n) od;
# Alternatively with the P-transform (cf. A269941):
A269945_row := n -> PTrans(n, n->`if`(n=1, 1, 1/(n*(4*n-2))), (n, k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A269945_row(n)), n=0..8);
# Using the exponential generating function:
egf := 1 + t^2*(cosh(2*sinh(t*x/2)/t));
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, 2*(n-k+1)), k = 0..n):
seq(print(Trow(n)), n = 0..9);  # Peter Luschny, Feb 29 2024

T[n_, n_] = 1; T[n_ /; n >= 0, k_] /; 0 <= k < n := T[n, k] = T[n - 1, k - 1] + k^2*T[n - 1, k]; T[, ] = 0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Jean-François Alcover, Nov 27 2017 *)

# uses[PtransMatrix from A269941]
stirset2 = lambda n: 1 if n == 1 else 1/(n*(4*n-2))
norm = lambda n,k: (-1)^k*factorial(2*n)/factorial(2*k)
M = PtransMatrix(7, stirset2, norm)
for m in M: print(m)

T(n, k) = (-1)^k*((2*n)! / (2*k)!)*P[n, k](s(n)) where P is the P-transform and s(n) = 1/(n*(4*n-2)). The P-transform is defined in the link. Compare also the Sage and Maple implementations below.
T(n, 2) = (4^(n - 1) - 1)/3 for n >= 2 (cf. A002450).
T(n, n-1) = n*(n - 1)*(2*n - 1)/6 for n >= 1 (cf. A000330).
From Fabián Pereyra, Apr 25 2022: (Start)
T(n, k) = (1/(2*k)!)*Sum_{j=0..2*k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n).
T(n, k) = Sum_{j=2*k..2*n} (-k)^(2*n - j)*binomial(2*n, j)*Stirling2(j, 2*k).
T(n, k) = Sum_{j=0..2*n} (-1)^(j - k)*Stirling2(2*n - j, k)*Stirling2(j, k). (End)
T(n, k) = (2*n)! [t^(2*(n-k+1))] [x^(2*n)] (1 + t^2*(cosh(2*sinh(t*x/2)/t))). - Peter Luschny, Feb 29 2024

A345876 a(n) = Sum_{k=0..n} binomial(2n, n-k) k^n.

Original entry on oeis.org

1, 1, 8, 90, 1408, 28350, 697344, 20264244, 679313408, 25805186550, 1095482736640, 51397070440716, 2640925289349120, 147491783753286700, 8895880971425939456, 576279075821454657000, 39905347440408027725824, 2941534126495441574472870, 229966392623413457628168192

Offset: 0

Vaclav Kotesovec, Oct 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..360

Cf. A072034, A298851.

Join[{1}, Table[Sum[Binomial[2*n, n-k]*k^n, {k, 0, n}], {n, 1, 20}]]

a(n) = sum(k=0, n, binomial(2*n, n-k) * k^n); \\ Michel Marcus, Oct 03 2021

a(n) ~ 2^(2*n + 1/2) * r^(n+1) * n^n / (sqrt(1 + r^2) * exp(n) * (1 - r^2)^n), where r = 0.647918229029602749602061258113970414114660380467168496836586... is the positive root of the equation (1 + r) = (1 - r)*exp(1/r).

A299035 a(n) = [x^n] Product_{k=1..n} 1/(1-k^k*x).

Original entry on oeis.org

1, 1, 21, 23980, 4896624249, 327969374429859111, 11123496833223144303532943536, 273486179312859032380857823231575174373792, 6620886635410516590847876477644821623913997428738363459941

Offset: 0

Seiichi Manyama, Feb 01 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A002109.

Cf. A007820, A298851, A299036.

Table[SeriesCoefficient[Product[1/(1 - k^k*x), {k, 1, n}], {x, 0, n}], {n, 0, 10}] (* Vaclav Kotesovec, Feb 02 2018 *)

a(n) ~ n^(n^2). - Vaclav Kotesovec, Feb 02 2018

A299036 a(n) = [x^n] Product_{k=1..n} 1/(1-k!*x).

Original entry on oeis.org

1, 1, 7, 381, 502789, 33572762781, 175123095782787181, 99374457734129265819664221, 8158897372191288496224413025490409437, 124778468912108975502836576328262294089846582756189

Offset: 0

Seiichi Manyama, Feb 01 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..30

Cf. A000178, A036740.

Cf. A007820, A298851, A299035.

Table[SeriesCoefficient[Product[1/(1 - k!*x), {k, 1, n}], {x, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Feb 02 2018 *)

From Vaclav Kotesovec, Feb 02 2018: (Start)
a(n) ~ (n!)^n.
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12). (End)

A351800 a(n) = [x^n] 1/Product_{j=1..n} (1 - j^3*x).

Original entry on oeis.org

1, 1, 73, 28800, 33120201, 83648533275, 393764054984212, 3103381708489548640, 37965284782803741391413, 681476650259874114533077575, 17184647574689079046814198039765, 588057239856779143071625300022102376, 26548105106818292578525347802793561068860

Offset: 0

Alois P. Heinz, Feb 19 2022

nonn

			a(2) = (1*1)^3 + (1*2)^3 + (2*2)^3 = 1 + 8 + 64 = 73.

Alois P. Heinz, Table of n, a(n) for n = 0..149

Cf. A000442, A001700, A007820, A088218, A098436, A269948, A298851, A351804.

b:= proc(n, k) option remember; `if`(k=0, 1,
      add(b(j, k-1)*j^3, j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..15);

Table[SeriesCoefficient[Product[1/(1 - k^3*x), {k, 1, n}], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 17 2025 *)

a(n) = Sum_{p in {1..n}^n : p_i <= p_{i+1}} Product_{j=1..n} p_j^3.
a(n) = A098436(2n-1,n-1) = A269948(2n,n).
a(n) ~ c * d^n * n^(3*n - 1/2), where d = 1.54371040458513693750053812318801418996889528987425... and c = 0.71526493063554190404119140313248864511356727815244... - Vaclav Kotesovec, May 13 2025

A383853 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(4*n).

Original entry on oeis.org

1, 1, 260, 556032, 4641176128, 106519579045760, 5472276566891956224, 549375993583284180705280, 97867116732573493470161420288, 28783909470167571938915053763592192, 13216052972619446942074113385580542689280, 9058922175695195359062480694771506779050213376

Offset: 0

Vaclav Kotesovec, May 12 2025

nonn

Cf. A209289, A298851, A345876.

Join[{1}, Table[Sum[Binomial[2*n, k]*(n-k)^(4*n), {k, 0, n}], {n, 1, 12}]] (* or *)
Join[{1}, Table[Sum[Binomial[2*n, n+k]*k^(4*n), {k, 0, n}], {n, 1, 12}]]

a(n) = Sum_{k=0..n} binomial(2*n, n+k) * k^(4*n).

a(n) ~ 4^n * r^(4*n+1) * n^(4*n) / (sqrt(2 - r^2) * (1 - r^2)^n * exp(4*n)), where r = 0.9683644349844134852843167967986294187258222293516... is the root of the equation (1+r)/(1-r) = exp(4/r).

A384043 a(n) = [x^n] Product_{k=1..n} (1 + k^2x) / (1 - k^2x).

Original entry on oeis.org

1, 2, 50, 4188, 735600, 221302710, 101667388082, 66218673102680, 58048466179356672, 65901249246347377770, 94061755750395244537250, 164863945136411230998746612, 348110204753572939058548570000, 871547135491620353615820806025918, 2552918049709989779004770502542335650

Offset: 0

Vaclav Kotesovec, May 18 2025

nonn

Cf. A001044, A298851.

Cf. A350366, A384044, A351764.

Table[SeriesCoefficient[Product[(1+k^2*x)/(1-k^2*x), {k, 1, n}], {x, 0, n}], {n, 0, 15}]

a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 16.6871576653578743696262746377576281620174969944584774545888... and c = 0.1371163625236187865398447973928851799479072107076663329994...

A348081 a(n) = [x^n] Product_{k=1..2n} 1/(1 - k^2 x).

Original entry on oeis.org

1, 5, 627, 251498, 209609235, 298201326150, 646748606934510, 1986821811445598260, 8209989926930833199235, 43919039258570117113742270, 295300365118450495520630242042, 2437724587984574697761809904387340, 24239364659088896670563082403144467630

Offset: 0

Seiichi Manyama, Sep 27 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..180

Cf. A036969, A234324, A269945, A298851, A348082.

Table[SeriesCoefficient[Product[1/(1 - k^2*x), {k, 1, 2*n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2021 *)

a(n) = polcoef(1/prod(k=1, 2*n, 1-k^2*x+x*O(x^n)), n);

a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 78.52705817932973261726305432915417900827309581709564977985583533852704254... = (2+r)^6 / (r^2*(4+r)^2), where r = 0.329482909104375658581668801636329590897344... is the root of the equation 4+r = r*exp(6/(2+r)) and c = (2+r)/(Pi^(3/2)*sqrt(32 - 4*r*(4+r))) = 0.0815842039686253664272939415761688591712635596695065951780203519... - Vaclav Kotesovec, Oct 16 2021, updated May 17 2025
From Seiichi Manyama, May 13 2025: (Start)
a(n) = A036969(3*n,2*n) = A269945(3*n,2*n).
a(n) = (1/(4*n)!) * Sum_{k=0..4*n} (-1)^k * (2*n-k)^(6*n) * binomial(4*n,k).
a(n) = Sum_{k=0..2*n} (-2*n)^k * binomial(6*n,k) * Stirling2(6*n-k,4*n).
a(n) = Sum_{k=0..2*n} (-1)^k * Stirling2(2*n+k,2*n) * Stirling2(4*n-k,2*n). (End)

cp's OEIS Frontend

A350366 a(n) = [x^n] Product_{k=1..n} (1 + k*x)/(1 - k*x).

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A350376 a(n) = [x^n] Product_{k=1..n} 1/(1 - k*x)^2.

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A269945 Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.

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A345876 a(n) = Sum_{k=0..n} binomial(2*n, n-k) * k^n.

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A299035 a(n) = [x^n] Product_{k=1..n} 1/(1-k^k*x).

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Mathematica

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A299036 a(n) = [x^n] Product_{k=1..n} 1/(1-k!*x).

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Mathematica

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A351800 a(n) = [x^n] 1/Product_{j=1..n} (1 - j^3*x).

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A383853 a(n) = Sum_{k=0..n} binomial(2*n, k) * (n-k)^(4*n).

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A350366 a(n) = [x^n] Product_{k=1..n} (1 + kx)/(1 - kx).

A345876 a(n) = Sum_{k=0..n} binomial(2n, n-k) k^n.

A383853 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(4*n).

A384043 a(n) = [x^n] Product_{k=1..n} (1 + k^2x) / (1 - k^2x).

A348081 a(n) = [x^n] Product_{k=1..2n} 1/(1 - k^2 x).