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%I #6 Dec 06 2013 01:29:35
%S 1,1,6,150,15684,6626832,11412679110,80341130055678,
%T 2305199459532741522,268629428492391824756106,
%U 126762373497858122449971372498,241676422998164497873224935953948770,1858392533076949187099229893507827126982592,57560655711123829878000426546315591572901023820252
%N G.f. A(x) satisfies: the sum of the coefficients of x^k, k=0..n, in A(x)^n equals 2^(n^2) for n>=0.
%H Paul D. Hanna, <a href="/A232689/b232689.txt">Table of n, a(n) for n = 0..40</a>
%F Given g.f. A(x), Sum_{k=0..n} [x^k] A(x)^n = 2^(n^2).
%e G.f.: A(x) = 1 + x + 6*x^2 + 150*x^3 + 15684*x^4 + 6626832*x^5 +...
%e ILLUSTRATION OF INITIAL TERMS.
%e If we form an array of coefficients of x^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], 6, 150, 15684, 6626832, 11412679110, 80341130055678, ...;
%e A^2: [1, 2, 13], 312, 31704, 13286832, 22838822592, 160705169696760, ...;
%e A^3: [1, 3, 21, 487], 48078, 19980558, 34278483114, 241092139452066, ...;
%e A^4: [1, 4, 30, 676, 64825], 26708592, 45731714160, 321502059924816, ...;
%e A^5: [1, 5, 40, 880, 81965, 33471541], 57198570060, 401934951793740, ...;
%e A^6: [1, 6, 51, 1100, 99519, 40270038, 68679106021], 482390835814224, ...;
%e A^7: [1, 7, 63, 1337, 117509, 47104743, 80173378159, 562869732819493], ...; ...
%e then the sum of the coefficients of x^k, k=0..n, in A(x)^n (shown above in brackets) equals 2^(n^2):
%e 2^0 = 1 = 1;
%e 2^1 = 1 + 1 = 2;
%e 2^4 = 1 + 2 + 13 = 16;
%e 2^9 = 1 + 3 + 21 + 487 = 512;
%e 2^16 = 1 + 4 + 30 + 676 + 64825 = 65536;
%e 2^25 = 1 + 5 + 40 + 880 + 81965 + 33471541 = 33554432;
%e 2^36 = 1 + 6 + 51 + 1100 + 99519 + 40270038 + 68679106021 = 68719476736; ...
%o (PARI) /* By Definition (slow): */
%o {a(n)=if(n==0, 1, (2^(n^2) - sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j)^n + x*O(x^k), k)))/n)}
%o for(n=0, 20, print1(a(n)*1!, ", "))
%o (PARI) /* Faster, using series reversion: */
%o {a(n)=local(B=sum(k=0, n+1, 2^(k^2)*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)); polcoeff(x/serreverse(x*G), n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A232687, A232606, A002416.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 06 2013
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