A260532 -id:A260532 - cp's OEIS frontend

A051295 a(0)=1; thereafter, a(m+1) = Sum_{k=0..m} k!*a(m-k).

1, 1, 2, 5, 15, 54, 235, 1237, 7790, 57581, 489231, 4690254, 49986715, 585372877, 7463687750, 102854072045, 1522671988215, 24093282856182, 405692082526075, 7242076686885157, 136599856992122366, 2714409550073698925, 56674981258436882463, 1240409916125255533662

Offset: 0

Leroy Quet

a(n) = number of permutations on [n] that contain a 132 pattern only as part of a 4132 pattern. For example, a(4) = 15 counts the 14 132-avoiding permutations on [4] (Catalan numbers A000108) and 4132.
a(n) is the number of permutations on [n] that contain a (scattered) 342 pattern only as part of a 1342 pattern. For example, 412635 fails because 463 is an offending 342 pattern (= 231 pattern).
This sequence gives the number of permutations of {1,2,...,n} such that the elements of each cycle of the permutation form an interval. - Michael Albert, Dec 14 2004
Starting (1, 2, 5, 15, ...) = row sums of triangle A143965. - Gary W. Adamson, Apr 10 2009
Number of compositions of n where there are (k-1)! sorts of part k. - Joerg Arndt, Aug 04 2014

			a[ 4 ]=15=a[ 3 ]*0!+a[ 2 ]*1!+a[ 1 ]*2!+a[ 0 ]*3!=5*1+2*1+1*2+1*6.
As to matrix M, a(3) = 5 since the top row of M^n = (5, 5, 4, 1), with a(4) = 15 = (5 + 5 + 4 + 1).

Vincenzo Librandi and Vaclav Kotesovec, Table of n, a(n) for n = 0..448 (first 100 terms from Vincenzo Librandi)
David Callan, A combinatorial interpretation of the eigensequence for composition, arXiv:math/0507169 [math.CO], 2005.
David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.
Stefan Forcey, Aaron Lauve and Frank Sottile, Cofree compositions of coalgebras, Annals of Combinatorics 17 (1) pp. 105-130 March, 2013.
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
Lara Pudwell, Enumeration Schemes for Permutations Avoiding Barred Patterns, El. J. Combinat. 17 (1) (2010) R29.

Cf. A051296, A260532.
Row sums of A084938.
Cf. A143965. - Gary W. Adamson, Apr 10 2009
Column k=0 of A381529.

a := proc(n) option remember; `if`(n<2, 1, add(a(n-j-1)*j!, j=0..n-1)) end proc: seq(a(n), n=0..30); # Vaclav Kotesovec, Jul 28 2015

Table[Coefficient[Series[E^x/(E^x-ExpIntegralEi[x]),{x,Infinity,20}],x,-n],{n,0,20}] (* Vaclav Kotesovec, Feb 22 2014 *)

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=(1+x^2*deriv(A)/A)/(1-x));polcoeff(A,n)} \\ Paul D. Hanna, Aug 02 2008

It appears that the INVERT transform of factorial numbers A000142 gives 1, 2, 5, 15, 54, 235, 1237, ... - Antti Karttunen, May 30 2003
This is true: translating the defining recurrence to a generating function identity yields A(x) = 1/(1 - (0!*x + 1!*x^2 + 2!*x^3 + ...)) which is the INVERT formula.
In other words: let F(x) = Sum_{n>=0} n!*x^n then the g.f. is 1/(1-x*F(x)), cf. A052186 (g.f. F(x)/(1+x*F(x))). - Joerg Arndt, Apr 25 2011
a(n) = Sum_{k>=0} A084938(n, k). - Philippe Deléham, Feb 05 2004
G.f. A(x) satisfies: A(x) = (1-x)*A(x)^2 - x^2*A'(x). - Paul D. Hanna, Aug 02 2008
G.f.: A(x) = 1/(1-x/(1-1*x/(1-1*x/(1-2*x/(1-2*x/(1-3*x/(1-3*x...))))))) (continued fraction). - Paul Barry, Sep 25 2008
From Gary W. Adamson, Jul 22 2011: (Start)
a(n) = upper left term in M^n, M = an infinite square production matrix in which a column of 1's is prepended to Pascal's triangle, as follows:
  1, 1, 0, 0, 0, ...
  1, 1, 1, 0, 0, ...
  1, 1, 2, 1, 0, ...
  1, 1, 3, 3, 1, ...
  ...
Also, a(n+1) = sum of top row terms of M^n. (End)
G.f.: 1+x/(U(0)-x) where U(k) = 1 + x*k - x*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012
G.f.: 1/(U(0) - x) where U(k) = 1 - x*(k+1)/(1 - x*(k+1)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Nov 12 2012
a(n) ~ (n-1)! * (1 + 2/n + 7/n^2 + 31/n^3 + 165/n^4 + 1025/n^5 + 7310/n^6 + 59284/n^7 + 543702/n^8 + 5618267/n^9 + 65200918/n^10), for coefficients see A260532. - Vaclav Kotesovec, Jul 28 2015

More terms from Vincenzo Librandi, Feb 23 2013

A260503 Coefficients in an asymptotic expansion of sequence A003319.

Original entry on oeis.org

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

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)).

A260530 Coefficients in asymptotic expansion of sequence A051296.

Original entry on oeis.org

1, 2, 7, 35, 216, 1575, 13243, 126508, 1359437, 16312915, 217277446, 3194459333, 51557948291, 908431129702, 17376289236947, 358847480175063, 7959468559605624, 188702262366570387, 4760773506835189975, 127312428854513811012, 3596091234340397964321

Offset: 0

Vaclav Kotesovec, Jul 28 2015

nonn

			A051296(n) / n! ~ 1 + 2/n + 7/n^2 + 35/n^3 + 216/n^4 + 1575/n^5 + 13243/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. A051296, A256168, A260491, A260503, A260532, A260578.

nmax = 30; b = CoefficientList[Assuming[Element[x, Reals], Series[E^(2/x)*x^2 / (ExpIntegralEi[1/x] - 2*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! / (2 * (log(2))^(k+1)).

A134378 A084938 * [1,2,3,...], where A084938 is taken as lower triangular matrix.

Original entry on oeis.org

1, 2, 5, 14, 44, 158, 663, 3310, 19759, 139660, 1147120, 10729684, 112309193, 1297522650, 16371057801, 223716758346, 3289199827236, 51745234494858, 867023125576027, 15411557297930534, 289610871340870883, 5736017561257017128, 119413599371241577016

Offset: 0

Gary W. Adamson, Oct 22 2007, corrected Oct 26 2007

nonn

			a(4) = 44 = (0, 6, 5, 3, 1) dot (1, 2, 3, 4, 5) = (0 + 12 + 15 + 12 + 5).

Alois P. Heinz, Table of n, a(n) for n = 0..170

Cf. A084938, A260532.

series((1-x*hypergeom([1,1],[],x))^(-2), x=0, 50);  # appears to generate the sequence - Mark van Hoeij, Apr 22 2013

CoefficientList[Series[1/(1-x*HypergeometricPFQ[{1, 1}, {}, x])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 17 2015 after Mark van Hoeij *)
CoefficientList[Assuming[Element[x,Reals], Series[E^(2/x)/(ExpIntegralEi[1/x]-E^(1/x))^2, {x,0,25}]],x] (* Vaclav Kotesovec, Aug 03 2015 *)

a(n) ~ 2 * (n-1)! * (1 + 3/n + 12/n^2 + 58/n^3 + 327/n^4 + 2107/n^5 + 15329/n^6 + 125041/n^7 + 1139467/n^8 + 11582187/n^9 + 131230827/n^10). - Vaclav Kotesovec, Mar 17 2015

More terms from Alois P. Heinz, Apr 27 2012

A305275 Coefficients in asymptotic expansion of sequence A302557.

Original entry on oeis.org

1, 0, 2, 6, 35, 256, 2187, 21620, 243947, 3098528, 43799819, 682540780, 11630529643, 215190967544, 4296657514283, 92083313483300, 2108244638675035, 51350077108834832, 1325682930813985547, 36157047428501464220, 1038793351537388253211, 31354977545074731373512

Offset: 0

Vaclav Kotesovec, Aug 18 2018

nonn

			A302557(n) / (exp(-1) * n!) ~ 1 + 2/n^2 + 6/n^3 + 35/n^4 + 256/n^5 + 2187/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..75

Cf. A256168, A260491, A260503, A260530, A260532, A260578, A260948, A260949, A260950, A302557.

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

cp's OEIS Frontend

A051295 a(0)=1; thereafter, a(m+1) = Sum_{k=0..m} k!*a(m-k).

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A260503 Coefficients in an asymptotic expansion of sequence A003319.

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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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A260530 Coefficients in asymptotic expansion of sequence A051296.

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A134378 A084938 * [1,2,3,...], where A084938 is taken as lower triangular matrix.

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