The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A187739 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A187739

G.f.: Sum_{n>=0} (3*n+2)^n * x^n / (1 + (3*n+2)*x)^n.
(history; published version)

#14 by Paul D. Hanna at Thu Jan 03 01:51:04 EST 2013

STATUS

editing

approved

#13 by Paul D. Hanna at Thu Jan 03 01:51:01 EST 2013

COMMENTS

More generally,

if Sum_{n>=0} a(n)*x^n = Sum_{n>=0} (b*n+c)^n * x^n / (1 + (b*n+c)*x)^n,

then Sum_{n>=0} a(n)*x^n/n! = (2 - 2*(b-c)*x + b*(b-2*c)*x^2)/(2*(1-b*x)^2)

so that a(n) = (b*n + (b+2*c)) * b^(n-1) * n!/2 for n>0 with a(0)=1.

STATUS

approved

editing

#12 by Paul D. Hanna at Thu Jan 03 01:33:20 EST 2013

STATUS

editing

approved

#11 by Paul D. Hanna at Thu Jan 03 01:33:17 EST 2013

CROSSREFS

Cf. A187735, A014479, A187738, A221160, A221161, A187740.

STATUS

approved

editing

#10 by Paul D. Hanna at Thu Jan 03 01:17:15 EST 2013

STATUS

editing

approved

#9 by Paul D. Hanna at Thu Jan 03 01:17:12 EST 2013

CROSSREFS

Cf. A187735, A014479, A187738, A221160, A221161.

STATUS

approved

editing

#8 by Paul D. Hanna at Thu Jan 03 01:03:30 EST 2013

STATUS

editing

approved

#7 by Paul D. Hanna at Thu Jan 03 01:03:26 EST 2013

NAME

G.f.: Sum_{n>=0} (3*n+2)^n * x^n / (1 + (3*n+2)*x)^n.

FORMULA

a(n) = (3*n+7) * 3^(n-1) * n!/2 for n>0 with a(0)=1.

EXAMPLE

G.f.: A(x) = 1 + 45*x + 3339*x^2 + 378432*x^3 + 55086156*x^4 + 97200106920*x^5 +...

PROG

(PARI) {a(n)=polcoeff(sum(m=0, n, ((3*m+2)*x)^m/(1+(3*m+2)*x +x*O(x^n))^m), n)}

CROSSREFS

Cf. A187735, A187738.

STATUS

approved

editing

#6 by Paul D. Hanna at Thu Jan 03 00:50:30 EST 2013

STATUS

editing

approved

#5 by Paul D. Hanna at Thu Jan 03 00:50:26 EST 2013

NAME

allocated for Paul D. Hanna

G.f.: Sum_{n>=0} (3*n+2)^n * x^n / (1 + (3*n+2)*x)^n.

DATA

1, 5, 39, 432, 6156, 106920, 2187000, 51438240, 1366787520, 40474546560, 1321374902400, 47140942464000, 1824354473356800, 76113765702374400, 3405263691641011200, 162618715070203392000, 8256027072794941440000, 444024146933226123264000, 25217509310311152586752000

OFFSET

0,2

FORMULA

a(n) = (3*n+7) * 3^(n-1) * n!/2 for n>0 with a(0)=1.

E.g.f.: (2 - 2*x - 3*x^2) / (2*(1-3*x)^2).

EXAMPLE

G.f.: A(x) = 1 + 4*x + 33*x^2 + 378*x^3 + 5508*x^4 + 97200*x^5 +...

where

A(x) = 1 + 5*x/(1+5*x) + 8^2*x^2/(1+8*x)^2 + 11^3*x^3/(1+11*x)^3 + 14^4*x^4/(1+14*x)^4 + 17^5*x^5/(1+17*x)^5 +...

PROG

(PARI) {a(n)=polcoeff(sum(m=0, n, ((3*m+2)*x)^m/(1+(3*m+2)*x +x*O(x^n))^m), n)}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A187735, A187738.

KEYWORD

allocated

nonn

AUTHOR

Paul D. Hanna, Jan 03 2013

STATUS

approved

editing