A206155 -id:A206155

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A206156

a(n) = Sum_{k=0..n} binomial(n,k)^(2*k).

+10
4

1, 2, 6, 92, 5410, 1400652, 2687407464, 18947436116184, 536104663173431874, 130559883231879141946580, 136031455187223511721647272376, 483565526783420050082035900177878504, 14487924180895151383693101563813954330590756

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Ignoring initial term a(0), equals the logarithmic derivative of A206155.

LINKS

Table of n, a(n) for n=0..12.

FORMULA

Limit n->infinity a(n)^(1/n^2) = r^(2*r^2/(1-2*r)) = 2.3520150420944489879258119..., where r = 0.70350607643066243... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Mar 03 2014

EXAMPLE

L.g.f.: L(x) = 2*x + 6*x^2/2 + 92*x^3/3 + 5410*x^4/4 + 1400652*x^5/5 +...

where exponentiation yields A206155:

exp(L(x)) = 1 + 2*x + 5*x^2 + 38*x^3 + 1425*x^4 + 283002*x^5 + 448468978*x^6 +...

Illustration of initial terms:

a(1) = 1^0 + 1^2 = 2;

a(2) = 1^0 + 2^2 + 1^4 = 6;

a(3) = 1^0 + 3^2 + 3^4 + 1^6 = 92;

a(4) = 1^0 + 4^2 + 6^4 + 4^6 + 1^8 = 5410;

a(5) = 1^0 + 5^2 + 10^4 + 10^6 + 5^8 + 1^10 = 1400652; ...

MATHEMATICA

Table[Sum[Binomial[n, k]^(2*k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)

PROG

(PARI) {a(n)=sum(k=0, n, binomial(n, k)^(2*k))}

for(n=0, 16, print1(a(n), ", "))

CROSSREFS

Cf. A206155 (exp), A184731, A206154, A206158, A206152, A220359.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 04 2012

STATUS

approved

A206153

G.f.: exp( Sum_{n>=1} A206154(n)*x^n/n ), where A206154(n) = Sum_{k=0..n} binomial(n,k)^(k+2).

+10
3

1, 2, 7, 48, 693, 26632, 2542514, 533442978, 278979307990, 343728261289376, 904762216681139381, 5771110378770242683658, 88742047516327429085056353, 2912737209806573079629325613400, 224604736339682169442980060945290802

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Logarithmic derivative yields A206154.

LINKS

Table of n, a(n) for n=0..14.

EXAMPLE

G.f.: A(x) = 1 + 2*x + 7*x^2 + 48*x^3 + 693*x^4 + 26632*x^5 + 2542514*x^6 +...

where the logarithm of the g.f. begins:

log(A(x)) = 2*x + 10*x^2/2 + 110*x^3/3 + 2386*x^4/4 + 125752*x^5/5 + 14921404*x^6/6 +...+ A206154(n)*x^n/n +...

PROG

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

for(n=0, 16, print1(a(n), ", "))

CROSSREFS

Cf. A206154 (log), A184730, A206155, A206157, A206151.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 04 2012

STATUS

approved

A206157

G.f.: exp( Sum_{n>=1} A206158(n)*x^n/n ), where A206158(n) = Sum_{k=0..n} binomial(n,k)^(2*k+1).

+10
3

1, 2, 7, 102, 6261, 2423430, 6686021554, 61335432894584, 2941073857435300366, 1190520035262419577871332, 1696475310227140760623646031573, 9980324833243234634513255755001535870, 565171444566758371735408026461987217216896790

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Logarithmic derivative yields A206158.

LINKS

Table of n, a(n) for n=0..12.

EXAMPLE

G.f.: A(x) = 1 + 2*x + 7*x^2 + 102*x^3 + 6261*x^4 + 2423430*x^5 +...

where the logarithm of the g.f. begins:

log(A(x)) = 2*x + 10*x^2/2 + 272*x^3/3 + 24226*x^4/4 + 12053252*x^5/5 + 40086916024*x^6/6 +...+ A206158(n)*x^n/n +...

PROG

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

for(n=0, 16, print1(a(n), ", "))

CROSSREFS

Cf. A206158 (log), A184730, A206153, A206155, A206151.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 04 2012

STATUS

approved

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