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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.)