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A111529 Row 2 of table A111528. 15
1, 1, 4, 22, 148, 1156, 10192, 99688, 1069168, 12468208, 157071424, 2126386912, 30797423680, 475378906432, 7793485765888, 135284756985472, 2479535560687360, 47860569736036096, 970606394944476160, 20635652201785613824, 459015456156148876288, 10662527360021306782720 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Robert Israel, Table of n, a(n) for n = 0..410

Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.

Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

FORMULA

G.f.: (1/2)*log(sum_{n>=0} (n+1)!*x^n) = sum_{n>=1} a(n)*x^n/n.

G.f.: 1/(1+2*x - 3*x/(1+3*x - 4*x/(1+4*x -... (continued fraction).

a(n) = sum_{k,0<=k<=n} 2^(n-k)*A089949(n,k). - Philippe Deléham, Oct 16 2006

G.f. 1/(2*x-G(0)) where G(k)=  2*x - 1 - k*x - x*(k+1)/G(k+1); G(0)=x (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 14 2012

G.f.: 1/(2*x) - 1/(G(0) - 1) where G(k)= 1 + x*(k+1)/(1 - 1/(1 + 1/G(k+1)));(continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 20 2012

G.f.: 1 + x/(G(0)-2*x) where G(k) = 1 + (k+1)*x - x*(k+3)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2012

G.f.: (1 + 1/Q(0))/2, where Q(k)= 1 + k*x - x*(k+2)/Q(k+1); (continued fraction). In general, the g.f. for row (r+2) is (r+1 + 1/Q(0))/(r+2). - Sergei N. Gladkovskii, May 04 2013

G.f.: W(0), where W(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+3)/( x*(k+3) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013

a(n) ~ n! * n^2/2 * (1 - 1/n - 2/n^2 - 8/n^3 - 52/n^4 - 436/n^5 - 4404/n^6 - 51572/n^7 - 683428/n^8 - 10080068/n^9 - 163471284/n^10), where the coefficients are given by (n+2)*(n+1)/n^2 * Sum_{k>=0} A260491(k)/(n+2)^k. - Vaclav Kotesovec, Jul 27 2015

a(n) = -A077607(n+2)/2. - Vaclav Kotesovec, Jul 29 2015

EXAMPLE

(1/2)*log(1 + 2*x + 6*x^2 + ... + (n+1)!/1!*x^n + ...)

= x + 4/2*x^2 + 22/3*x^3 + 148/4*x^4 + 1156/5*x^5 + ...

MAPLE

N:= 30: # to get a(0) to a(N)

g:= 1/2*log(add((n+1)!*x^n, n=0..N+1)):

S:= series(g, x, N+1);

1, seq(j*coeff(S, x, j), j=0..N); # Robert Israel, Jul 10 2015

MATHEMATICA

T[n_, k_] := T[n, k] = Which[n<0 || k<0, 0, k==0 || k==1, 1, n==0, k!, True, (T[n-1, k+1]-T[n-1, k])/n - Sum[T[n, j] T[n-1, k-j], {j, 1, k-1}]];

a[n_] := T[2, n];

Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 09 2018 *)

PROG

(PARI) {a(n)=if(n<0, 0, if(n==0, 1, (n/2)*polcoeff(log(sum(m=0, n, (m+1)!/1!*x^m)), n)))}

CROSSREFS

Cf. A111528 (table), A003319 (row 1), A111530 (row 3), A111531 (row 4), A111532 (row 5), A111533 (row 6), A111534 (diagonal).

Cf. A077607, A089949, A260491.

Sequence in context: A199418 A112898 A253095 * A228883 A307439 A189845

Adjacent sequences:  A111526 A111527 A111528 * A111530 A111531 A111532

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 06 2005

STATUS

approved

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Last modified December 15 09:05 EST 2019. Contains 329995 sequences. (Running on oeis4.)