|
%I #11 Oct 20 2020 04:00:34
%S 1,1,5,67,1937,98791,7744549,857382695,126889656641,24157912257775,
%T 5749369223697701,1672527291075462559,584038879457972531185,
%U 241150002566590866157943,116245385996298375640197893,64707252902905394310560934391,41198982747438307655532993553409
%N 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.
%H Vaclav Kotesovec, <a href="/A244589/b244589.txt">Table of n, a(n) for n = 0..150</a>
%F E.g.f. A(x) satisfies: Sum_{k=0..n} [x^k] A(x)^n = (n+1)^n.
%F a(n) ~ exp(1-exp(-1)) * n! * n^(n-1). - _Vaclav Kotesovec_, Jul 03 2014
%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 67*x^3/3! + 1937*x^4/4! + 98791*x^5/5! +...
%e where
%e ILLUSTRATION OF INITIAL TERMS.
%e If we form an array of coefficients of x^k/k! in A(x)^n, n>=0, like so:
%e A^0: [1],0, 0, 0, 0, 0, 0, 0, ...;
%e A^1: [1, 1], 5, 67, 1937, 98791, 7744549, 857382695, ...;
%e A^2: [1, 2, 12], 164, 4560, 223652, 17054920, 1853019716, ...;
%e A^3: [1, 3, 21, 297], 8049, 380853, 28237293, 3008400909, ...;
%e A^4: [1, 4, 32, 472, 12608], 577864, 41657008, 4348646600, ...;
%e A^5: [1, 5, 45, 695, 18465, 823475], 57747565, 5903103995, ...;
%e A^6: [1, 6, 60, 972, 25872, 1127916, 77020344], 7706019180, ...;
%e A^7: [1, 7, 77, 1309, 35105, 1502977, 100075045, 9797289761], ...; ...
%e 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:
%e 1^0 = 1;
%e 2^1 = 1 + 1 = 2;
%e 3^2 = 1 + 2 + 12/2! = 9;
%e 4^3 = 1 + 3 + 21/2! + 297/3! = 64;
%e 5^4 = 1 + 4 + 32/2! + 472/3! + 12608/4! = 625;
%e 6^5 = 1 + 5 + 45/2! + 695/3! + 18465/4! + 823475/5! = 7776;
%e 7^6 = 1 + 6 + 60/2! + 972/3! + 25872/4! + 1127916/5! + 77020344/6! = 117649; ...
%o (PARI) /* By Definition (slow): */
%o {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)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) /* Faster, using series reversion: */
%o {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));
%o for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); n!*polcoeff(x/serreverse(x*G), n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A244577, A263075.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 30 2014
|
