A007820 -id:A007820 - cp's OEIS frontend

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

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)

A327416 a(n) = A156289(2*n, n).

Original entry on oeis.org

1, 1, 63, 21120, 20585565, 44025570225, 175418438510700, 1169944052730453000, 12110024900113702687125, 183906442861089163922581875, 3923248989041777334572795737575, 113570018319217734510803494872644700, 4337118170117525113961286942555563803500

Offset: 0

Peter Luschny, Sep 14 2019

nonn

Cf. A007820 (m=1), this sequence (m=2), A327417 (m=3), A327418 (m=4).

Associated triangles: A048993 (m=1), A156289 (m=2), A291451 (m=3), A291452 (m=4).

@cached_function
def P(m, n):
    x = polygen(ZZ)
    if n == 0: return x^0
    return expand(sum(binomial(m*n, m*k)*P(m, n-k)*x for k in (1..n)))
def A327416(n): return P(2, 2*n).list()[n]//factorial(n)
print([A327416(n) for n in range(13)])

A327417 a(n) = A291451(2*n, n).

Original entry on oeis.org

1, 1, 682, 7128576, 429120851544, 94066556834970720, 57496301859366489159040, 82247725949165261902606309120, 243263294602173417290925789755652480, 1356449073308047884259226117174893156252800, 13275987570857688650109290727617026478737341900800

Offset: 0

Peter Luschny, Sep 14 2019

nonn

Cf. A007820 (m=1), A327416 (m=2), this sequence (m=3), A327418 (m=4).

Associated triangles: A048993 (m=1), A156289 (m=2), A291451 (m=3), A291452 (m=4).

# uses[P from A327416]
def A327417(n): return P(3, 2*n).list()[n]//factorial(n)
print([A327417(n) for n in range(11)])

A327418 a(n) = A291452(2*n, n).

Original entry on oeis.org

1, 1, 8255, 2941884000, 11957867341948125, 294040106448733743008625, 30188472144950452369737153667500, 10143939867539251013312279527292897925000, 9389957475743686923255643914812959599614184703125, 21058194888200109612591474039339954750056969537259132421875

Offset: 0

Peter Luschny, Sep 14 2019

nonn

Cf. A007820 (m=1), A327416 (m=2), A327417 (m=3), this sequence (m=4).

Associated triangles: A048993 (m=1), A156289 (m=2), A291451 (m=3), A291452 (m=4).

# uses[P from A327416]
def A327418(n): return P(4, 2*n).list()[n]//factorial(n)
print([A327418(n) for n in range(10)])

A344445 Number of cycle-up-down permutations of [2n] having n cycles.

Original entry on oeis.org

1, 1, 7, 105, 2345, 69405, 2559667, 113073961, 5820788545, 342176336073, 22616620648895, 1660292619682697, 134029227728536985, 11800452870718122325, 1125324001129006580475, 115551341953019187183225, 12711056625162235880359425, 1491325482312555276046069905

Offset: 0

Alois P. Heinz, May 19 2021

nonn

For the definition of cycle-up-down permutations see A186366.

			a(2) = 7: (1)(243), (143)(2), (142)(3), (132)(4), (12)(34), (13)(24), (14)(23).

Alois P. Heinz, Table of n, a(n) for n = 0..338
Wikipedia, Permutation

Cf. A007820, A186366, A344532.

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
g:= proc(n) option remember; expand(`if`(n=0, 1,
      add(g(n-j)*binomial(n-1, j-1)*x*b(j-1, 0), j=1..n)))
    end:
a:= n-> coeff(g(2*n), x, n):
seq(a(n), n=0..18);

Join[{1}, Table[Sum[2^(2*n - 2*j + 1) * StirlingS1[2*j,n] * Sum[(-1)^k * k^(2*n) / ((j+k)!*(j-k)!), {k, 0, j}], {j, Floor[n/2], n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 22 2021 *)

a(n) = (2n)! * [x^(2n) y^n] 1/(1-sin(x))^y.
a(n) = A186366(2n,n).
a(n) ~ c * d^n * (n-1)!, where d = 7.3270710411718682766548233722838416956334898839746535623751... and c = 0.14278148012337362269164226210064788025688590260058738... - Vaclav Kotesovec, May 22 2021

A348087 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2k-1) x).

Original entry on oeis.org

1, 1, 13, 330, 12411, 618870, 38461522, 2863440580, 248440887123, 24616763946918, 2742625188929990, 339386813915985836, 46184075261030623710, 6854605372617955658940, 1101943692701420653738500, 190748265085183804327197000, 35373318817392757170821576835

Offset: 0

Seiichi Manyama, Sep 28 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..317

Cf. A001147, A007820, A039755, A348085, A348088.

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

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

a(n) = A039755(2*n-1,n-1) for n > 0.
a(n) = (1/((-2)^(n-1) * (n-1)!)) * Sum_{k=0..n-1} (-1)^k * (2*k+1)^(2*n-1) * binomial(n-1,k) for n > 0.
a(n) ~ 2^(3*n - 1) * n^(n - 1/2) / (sqrt(Pi*(1-c)) * (2-c)^n * c^(n - 1/2) * exp(n)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, Oct 02 2021
From Seiichi Manyama, May 16 2025: (Start)
a(n) = Sum_{k=0..n} 2^k * binomial(2*n-1,k+n-1) * Stirling2(k+n-1,n-1) for n > 0.
a(n) = Sum_{k=0..n} (-2)^k * (2*n-1)^(n-k) * binomial(2*n-1,k+n-1) * Stirling2(k+n-1,n-1) for n > 0. (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

A383881 a(n) = [x^n] Product_{k=1..3n} 1/(1 - kx).

Original entry on oeis.org

1, 6, 266, 22275, 2757118, 452329200, 92484925445, 22653141490980, 6466506598695390, 2108114165258886708, 772778072287000494520, 314641228029527540596455, 140880584836935832288402135, 68799366730032076856334789900, 36392216443342587869022660451080, 20728132932716479897744043460870000

Offset: 0

Vaclav Kotesovec, May 13 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A007820, A348084, A383882.

Cf. A217913.

[&+[Abs(StirlingSecond(4*n, 3*n))]: n in [0..15]]; // Vincenzo Librandi, May 21 2025

Table[SeriesCoefficient[Product[1/(1-k*x), {k, 1, 3*n}], {x, 0, n}], {n, 0, 15}]
Table[StirlingS2[4*n, 3*n], {n, 0, 15}]
Table[SeriesCoefficient[(-1)^n/(Pochhammer[1 - 1/x, 3*n]*x^(3*n)), {x, 0, n}], {n, 0, 15}]

a(n) = Stirling2(4*n,3*n).

a(n) ~ (-1)^(3*n) * 4^(4*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w)) * exp(n) * 3^(3*n + 1/2) * w^(3*n) * (4/3 + w)^n), where w = LambertW(-4/(3*exp(4/3))).

A187661 Binomial convolution of the (signless) central Stirling numbers of the first kind and the central Stirling numbers of the second kind.

Original entry on oeis.org

1, 2, 20, 369, 10192, 379850, 17930697, 1027046517, 69216504576, 5363945384274, 469658243947850, 45827641349686636, 4928867833029014503, 579101340954599901152, 73778702335232336908585, 10129059530832922239925140

Offset: 0

Emanuele Munarini, Mar 12 2011

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A007820, A187646.

seq(sum(binomial(n,k) * abs(combinat[stirling1](2*k, k)) * combinat[stirling2](2*(n-k), n-k), k=0..n), n=0..12);

Table[Sum[Binomial[n, k]Abs[StirlingS1[2k, k]]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 15}]

makelist(sum(binomial(n,k)*abs(stirling1(2*k,k))*stirling2(2*n-2*k,n-k),k,0,n),n,0,12);

a(n) = sum(k=0, n, binomial(n,k)*abs(stirling(2*k, k, 1)*stirling(2*(n-k), n-k, 2))); \\ Michel Marcus, May 28 2017

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

a(n) ~ m * n^n * c^(2*n) * 2^(3*n-1) / (sqrt(Pi*(c-1)*n) * exp(n) * (2*c-1)^n), where c = -LambertW(-1,-exp(-1/2)/2) = 1.75643120862616967698..., and m = Sum_{j>=0} StirlingS2(2*j,j) * (2*c-1)^j / (j! * 2^(3*j) * c^(2*j)) = 1.170003674502655133465266152119563086693466... . - Vaclav Kotesovec, May 22 2014

cp's OEIS Frontend

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

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Mathematica

Formula

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

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Mathematica

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A327416 a(n) = A156289(2*n, n).

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SageMath

A327417 a(n) = A291451(2*n, n).

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Sage

A327418 a(n) = A291452(2*n, n).

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Sage

A344445 Number of cycle-up-down permutations of [2n] having n cycles.

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Maple

Mathematica

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A348087 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2*k-1) * x).

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PARI

PARI

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

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Maple

Mathematica

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A383881 a(n) = [x^n] Product_{k=1..3*n} 1/(1 - k*x).

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A348087 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2k-1) x).

A383881 a(n) = [x^n] Product_{k=1..3n} 1/(1 - kx).