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 A232689 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. 1
 1, 1, 6, 150, 15684, 6626832, 11412679110, 80341130055678, 2305199459532741522, 268629428492391824756106, 126762373497858122449971372498, 241676422998164497873224935953948770, 1858392533076949187099229893507827126982592, 57560655711123829878000426546315591572901023820252 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..40 FORMULA Given g.f. A(x), Sum_{k=0..n} [x^k] A(x)^n = 2^(n^2). EXAMPLE G.f.: A(x) = 1 + x + 6*x^2 + 150*x^3 + 15684*x^4 + 6626832*x^5 +... ILLUSTRATION OF INITIAL TERMS. If we form an array of coefficients of x^k in A(x)^n, n>=0, like so: A^0: [1],0,  0,    0,      0,         0,            0,                0, ...; A^1: [1, 1], 6,  150,  15684,   6626832,  11412679110,   80341130055678, ...; A^2: [1, 2, 13], 312,  31704,  13286832,  22838822592,  160705169696760, ...; A^3: [1, 3, 21,  487], 48078,  19980558,  34278483114,  241092139452066, ...; A^4: [1, 4, 30,  676,  64825], 26708592,  45731714160,  321502059924816, ...; A^5: [1, 5, 40,  880,  81965,  33471541], 57198570060,  401934951793740, ...; A^6: [1, 6, 51, 1100,  99519,  40270038,  68679106021], 482390835814224, ...; A^7: [1, 7, 63, 1337, 117509,  47104743,  80173378159,  562869732819493], ...; ... then the sum of the coefficients of x^k, k=0..n, in A(x)^n (shown above in brackets) equals 2^(n^2): 2^0  = 1 = 1; 2^1  = 1 + 1 = 2; 2^4  = 1 + 2 + 13 = 16; 2^9  = 1 + 3 + 21 +  487 = 512; 2^16 = 1 + 4 + 30 +  676 + 64825 = 65536; 2^25 = 1 + 5 + 40 +  880 + 81965 + 33471541 = 33554432; 2^36 = 1 + 6 + 51 + 1100 + 99519 + 40270038 + 68679106021 = 68719476736; ... PROG (PARI) /* By Definition (slow): */ {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)} for(n=0, 20, print1(a(n)*1!, ", ")) (PARI) /* Faster, using series reversion: */ {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)); for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); polcoeff(x/serreverse(x*G), n)} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A232687, A232606, A002416. Sequence in context: A188420 A089482 A126679 * A165436 A261066 A297737 Adjacent sequences:  A232686 A232687 A232688 * A232690 A232691 A232692 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 06 2013 STATUS approved

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Last modified April 23 03:51 EDT 2021. Contains 343199 sequences. (Running on oeis4.)