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A244589 E.g.f. A(x) satisfies the property that the sum of the coefficients of x^k, k=0..n, in A(x)^n equals (n+1)^n. 3
1, 1, 5, 67, 1937, 98791, 7744549, 857382695, 126889656641, 24157912257775, 5749369223697701, 1672527291075462559, 584038879457972531185, 241150002566590866157943, 116245385996298375640197893, 64707252902905394310560934391, 41198982747438307655532993553409 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..150

FORMULA

E.g.f. A(x) satisfies: Sum_{k=0..n} [x^k] A(x)^n = (n+1)^n.

a(n) ~ exp(1-exp(-1)) * n! * n^(n-1). - Vaclav Kotesovec, Jul 03 2014

EXAMPLE

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 67*x^3/3! + 1937*x^4/4! + 98791*x^5/5! +...

where

ILLUSTRATION OF INITIAL TERMS.

If we form an array of coefficients of x^k/k! in A(x)^n, n>=0, like so:

A^0: [1],0,  0,    0,     0,       0,         0,           0, ...;

A^1: [1, 1], 5,   67,  1937,   98791,   7744549,   857382695, ...;

A^2: [1, 2, 12], 164,  4560,  223652,  17054920,  1853019716, ...;

A^3: [1, 3, 21,  297], 8049,  380853,  28237293,  3008400909, ...;

A^4: [1, 4, 32,  472, 12608], 577864,  41657008,  4348646600, ...;

A^5: [1, 5, 45,  695, 18465,  823475], 57747565,  5903103995, ...;

A^6: [1, 6, 60,  972, 25872, 1127916,  77020344], 7706019180, ...;

A^7: [1, 7, 77, 1309, 35105, 1502977, 100075045,  9797289761], ...; ...

then we can illustrate how the sum of the coefficients of x^k, k=0..n, in A(x)^n (shown above in brackets) equals (n+1)^n:

1^0 = 1;

2^1 = 1 + 1 = 2;

3^2 = 1 + 2 + 12/2! = 9;

4^3 = 1 + 3 + 21/2! + 297/3! = 64;

5^4 = 1 + 4 + 32/2! + 472/3! + 12608/4! = 625;

6^5 = 1 + 5 + 45/2! + 695/3! + 18465/4! + 823475/5! = 7776;

7^6 = 1 + 6 + 60/2! + 972/3! + 25872/4! + 1127916/5! + 77020344/6! = 117649; ...

PROG

(PARI) /* By Definition (slow): */

{a(n)=if(n==0, 1, n!*((n+1)^n - sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j/j!)^n + x*O(x^k), k)))/n)}

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

(PARI) /* Faster, using series reversion: */

{a(n)=local(B=sum(k=0, n+1, (k+1)^k*x^k)+x^3*O(x^n), G=1+x*O(x^n));

for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); n!*polcoeff(x/serreverse(x*G), n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A244577, A263075.

Sequence in context: A323208 A262656 A293849 * A113064 A352860 A291807

Adjacent sequences:  A244586 A244587 A244588 * A244590 A244591 A244592

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 30 2014

STATUS

approved

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Last modified May 21 03:13 EDT 2022. Contains 353886 sequences. (Running on oeis4.)