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 A244577 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)!. 2
 1, 1, 2, 14, 196, 4652, 166168, 8232296, 535974416, 44186331248, 4489336764064, 550549455440096, 80153857492836928, 13665883723351362752, 2697370187692768024448, 610301579538939633684608, 156933087218604923576672512, 45515622704384079509089136384, 14789652457653705738777659937280 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..160 FORMULA Given g.f. A(x), Sum_{k=0..n} [x^k] A(x)^n = (n+1)!. a(n) ~ exp(-1) * (n!)^2. - Vaclav Kotesovec, Jul 03 2014 EXAMPLE E.g.f.: A(x) = 1 + x + 2*x^2/2! + 14*x^3/3! + 196*x^4/4! + 4652*x^5/5! +... 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,           0, ...; A^1: [1, 1], 2,  14,   196,   4652,  166168,   8232296,   535974416, ...; A^2: [1, 2,  6], 40,   528,  11824,  403840,  19373792,  1232259840, ...; A^3: [1, 3, 12,  84], 1068,  22716,  741456,  34375200,  2132407248, ...; A^4: [1, 4, 20, 152,  1912], 39008, 1218496,  54513152,  3292657664, ...; A^5: [1, 5, 30, 250,  3180,  62980],1889080,  81499400,  4785873360, ...; A^6: [1, 6, 42, 384,  5016,  97632, 2826288],117620256,  6706638336, ...; A^7: [1, 7, 56, 560,  7588, 146804, 4127200, 165911312], 9177810320, ...; A^8: [1, 8, 72, 784, 11088, 215296, 5918656, 230372480, 12358846848], ...; ... 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)!: 1! = 1; 2! = 1 + 1; 3! = 1 + 2 + 6/2!; 4! = 1 + 3 + 12/2! + 84/3!; 5! = 1 + 4 + 20/2! + 152/3! + 1912/4!; 6! = 1 + 5 + 30/2! + 250/3! + 3180/4! + 62980/5!; ... PROG (PARI) /* By Definition (slow): */ {a(n)=if(n==0, 1, n!*((n+1)! - 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)!*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. A244589, A232606, A232607, A232683, A232687. Sequence in context: A305112 A232686 A263766 * A090300 A213977 A322196 Adjacent sequences:  A244574 A244575 A244576 * A244578 A244579 A244580 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 30 2014 STATUS approved

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Last modified August 3 05:38 EDT 2020. Contains 336197 sequences. (Running on oeis4.)