A260530 -id:A260530 - cp's OEIS frontend

A260503 Coefficients in an asymptotic expansion of sequence A003319.

1, -2, -1, -5, -32, -253, -2381, -25912, -319339, -4388949, -66495386, -1100521327, -19751191053, -382062458174, -7924762051957, -175478462117633, -4132047373455024, -103115456926017761, -2718766185148876961, -75529218928863243200, -2205316818199975235447

Offset: 0

Vaclav Kotesovec, Jul 27 2015

sign

			A003319(n) / n! ~ 1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Cf. A003319, A259472, A261214, A261239, A261253, A261254.

Cf. A256168, A260491, A260530, A260532, A260578.

Flatten[{1, Table[Sum[Assuming[Element[x,Reals], SeriesCoefficient[E^(2/x)*x^2 / ExpIntegralEi[1/x]^2,{x,0,k}]] * StirlingS2[n-1,k-1], {k,1,n}], {n,1,20}]}] (* Vaclav Kotesovec, Aug 03 2015 *)

a(k) ~ -k! / (2 * (log(2))^(k+1)).
For n>0, Sum_{k=1..n} a(k) * Stirling1(n-1, k-1) = A259472(n). - Vaclav Kotesovec, Aug 03 2015
For n>0, a(n) = Sum_{k=1..n} A259472(k) * Stirling2(n-1, k-1). - Vaclav Kotesovec, Aug 03 2015

A051296 INVERT transform of factorial numbers.

Original entry on oeis.org

1, 1, 3, 11, 47, 231, 1303, 8431, 62391, 524495, 4960775, 52223775, 605595319, 7664578639, 105046841127, 1548880173119, 24434511267863, 410503693136559, 7315133279097607, 137787834979031839, 2734998201208351479, 57053644562104430735, 1247772806059088954855

Offset: 0

Leroy Quet

a(n) = Sum[ a1!a2!...ak! ] where (a1,a2,...,ak) ranges over all compositions of n. a(n) = number of trees on [0,n] rooted at 0, consisting entirely of filaments and such that the non-root labels on each filament, when arranged in order, form an interval of integers. A filament is a maximal path (directed away from the root) whose interior vertices all have outdegree 1 and which terminates at a leaf. For example with n=3, a(n) = 11 counts all n^(n-2) = 16 trees on [0,3] except the 3 trees {0->1, 1->2, 1->3}, {0->2, 2->1, 2->3}, {0->3, 3->1, 3->2} (they fail the all-filaments test) and the 2 trees {0->2, 0->3, 3->1}, {0->2, 0->1, 1->3} (they fail the interval-of-integers test). - David Callan, Oct 24 2004
a(n) is the number of lists of "unlabeled" permutations whose total length is n. "Unlabeled" means each permutation is on an initial segment of the positive integers (cf. A090238). Example: with dashes separating permutations, a(3) = 11 counts 123, 132, 213, 231, 312, 321, 1-12, 1-21, 12-1, 21-1, 1-1-1. - David Callan, Sep 20 2007
Number of compositions of n where there are k! sorts of part k. - Joerg Arndt, Aug 04 2014

			a(4) = 47 = 1*24 + 1*6 + 3*2 + 11*1.
a(4) = 47, the upper left term of M^4.

L. Comtet, Advanced Combinatorics, Reidel, 1974.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..440 (first 200 terms from Alois P. Heinz)
Jean-Paul Bultel, Ali Chouria, Jean-Gabriel Luque, and Olivier Mallet, Word symmetric functions and the Redfield-Polya theorem, hal-00793788, 2013.
Louis Comtet, Sur les coefficients de l'inverse de la série formelle Sum n! t^n, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.
Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Catalan-Spitzer permutations, arXiv:2310.06288 [math.CO], 2023. See p. 20.
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.

Cf. A051295, row sums of A090238.

Cf. A000142, A292778.

a:= proc(n) option remember; `if`(n<1, 1,
      add(a(n-i)*factorial(i), i=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 28 2015

CoefficientList[Series[Sum[Sum[k!*x^k, {k, 1, 20}]^n, {n, 0, 20}], {x, 0, 20}], x] (* Geoffrey Critzer, Mar 22 2009 *)

h = lambda x: 1/(1-x*hypergeometric((1, 2), (), x))
taylor(h(x),x,0,22).list() # Peter Luschny, Jul 28 2015

def A051296_list(len):
    R, C = [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n, 0, -1):
            C[k] = C[k-1] * k
        C[0] = sum(C[k] for k in (1..n))
        R.append(C[0])
    return R
print(A051296_list(23)) # Peter Luschny, Feb 21 2016

G.f.: 1/(1-Sum_{n>=1} n!*x^n).
a(0) = 1; a(n) = Sum_{k=1..n} a(n-k)*k! for n>0.
a(n) = Sum_{k>=0} A090238(n, k). - Philippe Deléham, Feb 05 2004
From Gary W. Adamson, Sep 26 2011: (Start)
a(n) is the upper left term of M^n, M = an infinite square production matrix as follows:
1, 1, 0, 0, 0, 0, ...
2, 0, 2, 0, 0, 0, ...
3, 0, 0, 3, 0, 0, ...
4, 0, 0, 0, 4, 0, ...
5, 0, 0, 0, 0, 5, ...
... (End)
G.f.: 1 + x/(G(0) - 2*x) where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 26 2012
a(n) ~ n! * (1 + 2/n + 7/n^2 + 35/n^3 + 216/n^4 + 1575/n^5 + 13243/n^6 + 126508/n^7 + 1359437/n^8 + 16312915/n^9 + 217277446/n^10), for coefficients see A260530. - Vaclav Kotesovec, Jul 28 2015
From Peter Bala, May 26 2017: (Start)
G.f. as an S-fraction: A(x) = 1/(1 - x/(1 - 2*x/(1 - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - n*x/(1 - (n - 1)*x/(1 - ...)))))))))). Cf. S-fraction for the o.g.f. of A000142.
A(x) = 1/(1 - x/(1 - x - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - ... ))))))))). (End)

Entry revised by David Callan, Sep 20 2007

A260578 Coefficients in asymptotic expansion of sequence A259869.

Original entry on oeis.org

1, 0, -2, -6, -29, -196, -1665, -16796, -194905, -2549468, -37055681, -592013436, -10307671769, -194225544124, -3937581243201, -85460277981116, -1977127315636969, -48573021658496348, -1262954975286604673, -34650561545808167292, -1000438355724912080873

Offset: 0

Vaclav Kotesovec, Jul 29 2015

sign

For k > 1 is a(k) negative.

			A259869(n) / (n!/exp(1)) ~ 1 - 2/n^2 - 6/n^3 - 29/n^4 - 196/n^5 - 1665/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..132
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A259869, A256168, A260491, A260503, A260530, A260532, A260948, A260949, A260950.

nmax = 20; b = CoefficientList[Assuming[Element[x, Reals], Series[x^2*E^(2 + 2/x)/ExpIntegralEi[1 + 1/x]^2, {x, 0, nmax}]], x]; Flatten[{1, Table[Sum[b[[k+1]]*StirlingS2[n-1, k-1], {k, 1, n}], {n, 1, nmax}]}] (* Vaclav Kotesovec, Aug 03 2015 *)

a(k) ~ -k! / (2 * exp(1) * (log(2))^(k+1)).

A256168 Coefficients in asymptotic expansion of sequence A052186.

Original entry on oeis.org

1, -2, 1, -1, -9, -59, -474, -4560, -50364, -625385, -8622658, -130751886, -2163331779, -38793751015, -749691306018, -15535914341831, -343749787006758, -8089725377931547, -201801866906374263, -5319643146604299835, -147774950436327236681

Offset: 0

Vaclav Kotesovec, Mar 17 2015

sign

For k > 2 is a(k) negative.

			A052186(n) / n! ~ 1 - 2/n + 1/n^2 - 1/n^3 - 9/n^4 - 59/n^5 - 474/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..394
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A052186, A260491, A260503, A260530, A260532, A260578.

nmax = 30; b = CoefficientList[Assuming[Element[x, Reals], Series[E^(2/x) / (ExpIntegralEi[1/x] + E^(1/x))^2, {x, 0, nmax}]], x]; Flatten[{1, Table[Sum[b[[k+1]]*StirlingS2[n-1, k-1], {k, 1, n}], {n, 1, nmax}]}] (* Vaclav Kotesovec, Aug 03 2015 *)

a(k) ~ -(k-1)! / (log(2))^k.

A260491 Coefficients in asymptotic expansion of sequence A077607.

Original entry on oeis.org

1, -4, 0, -8, -76, -752, -8460, -107520, -1522124, -23717424, -402941324, -7407988448, -146479479308, -3099229422352, -69863683041868, -1671667534710720, -42318672085310540, -1130167625049525232, -31758424368739424780, -936840101208573355680

Offset: 0

Vaclav Kotesovec, Jul 27 2015

sign

For k > 2 is a(k) negative.

			A077607(n) / (-n!) ~ 1 - 4/n - 8/n^3 - 76/n^4 - 752/n^5 - 8460/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..126
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A077607, A111529, A256168, A260503, A260530, A260532, A260578.

nmax = 30; b = CoefficientList[Assuming[Element[x, Reals], Series[x^4*E^(2/x)/(ExpIntegralEi[1/x] - x*E^(1/x))^2, {x, 0, nmax}]], x]; Flatten[{1, Table[Sum[b[[k+1]]*StirlingS2[n-1, k-1], {k, 1, n}], {n, 1, nmax}]}] (* Vaclav Kotesovec, Aug 03 2015 *)

a(k) ~ -k * k! / (4 * (log(2))^(k+2)).

A260532 Coefficients in asymptotic expansion of sequence A051295.

Original entry on oeis.org

1, 2, 7, 31, 165, 1025, 7310, 59284, 543702, 5618267, 65200918, 846462826, 12229783811, 195394019337, 3427472046792, 65526442181293, 1355785469986828, 30166624979467869, 717769036033944699, 18174105506247664633, 487655384740384445407, 13816406622559942660420

Offset: 0

Vaclav Kotesovec, Jul 28 2015

nonn

			A051295(n)/(n-1)! ~ 1 + 2/n + 7/n^2 + 31/n^3 + 165/n^4 + 1025/n^5 + 7310/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..134
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A051295, A134378, A256168, A260491, A260503, A260530, A260578.

nmax = 30; b = CoefficientList[Assuming[Element[x, Reals], Series[E^(2/x)*x / (ExpIntegralEi[1/x] - E^(1/x))^2, {x, 0, nmax+1}]], x]; Table[Sum[b[[k+1]] * StirlingS2[n, k-1], {k, 1, n+1}], {n, 0, nmax}] (* Vaclav Kotesovec, Aug 03 2015 *)

a(k) ~ 2 * (k-1)! / (log(2))^k.

a(n) = Sum_{k=0..n} A134378(k) * Stirling2(n, k). - Vaclav Kotesovec, Aug 04 2015

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A260503 Coefficients in an asymptotic expansion of sequence A003319.

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A051296 INVERT transform of factorial numbers.

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A260578 Coefficients in asymptotic expansion of sequence A259869.

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A256168 Coefficients in asymptotic expansion of sequence A052186.

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A260491 Coefficients in asymptotic expansion of sequence A077607.

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A260532 Coefficients in asymptotic expansion of sequence A051295.

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A305275 Coefficients in asymptotic expansion of sequence A302557.

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