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 A243942 O.g.f.: Sum_{n>=0} n^(2*n) * x^n / (1 - n*x)^n * exp( -n^2*x / (1 - n*x) ) / n!. 2
 1, 1, 8, 121, 2698, 79654, 2929238, 129004633, 6619919386, 387904397222, 25555935470016, 1869945551975658, 150459006927310348, 13203459856456213172, 1254972882696473807298, 128439184335788533011489, 14082139161229781077548346, 1646731810035799151750487814 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Table of n, a(n) for n=0..17. FORMULA a(n) = Sum_{k=1..n} C(n-1,k-1) * Stirling2(n+k,k) for n>0, a(0)=1. a(n) = c * (r^2/((1-r)*(2*r-1)))^n * n^(n-1/2) / exp(n), where r = 0.859294411517830517100430385442711799997876163... is the root of the equation (1-r)*(1+r)/r^2 = -LambertW(-exp(-1-1/r)*(1+r)/r), and c = 0.4180257159270405799046057130547446708890452... . - Vaclav Kotesovec, Aug 11 2014 EXAMPLE O.g.f.: A(x) = 1 + x + 8*x^2 + 121*x^3 + 2698*x^4 + 79654*x^5 + 2929238*x^6 +... where A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^4*x^2/(1-2*x)^2*exp(-4*x/(1-2*x))/2! + 3^6*x^3/(1-3*x)^3*exp(-9*x/(1-3*x))/3! + 4^8*x^4/(1-4*x)^4*exp(-16*x/(1-4*x))/4! + 5^10*x^5/(1-5*x)^5*exp(-25*x/(1-5*x))/5! +... simplifies to a power series in x with integer coefficients. Illustrate the terms by: a(1) = 1*1 = 1; a(2) = 1*1 + 1*7 = 8; a(3) = 1*1 + 2*15 + 1*90 = 121; a(4) = 1*1 + 3*31 + 3*301 + 1*1701 = 2698; a(5) = 1*1 + 4*63 + 6*966 + 4*7770 + 1*42525 = 79654; ... where Stirling2(n+k,k) forms a rectangular table as follows: 1, 1, 1, 1, 1, 1, 1, 1, ...; 0, 1, 3, 6, 10, 15, 21, 28, ...; 0, 1, 7, 25, 65, 140, 266, 462, ...; 0, 1, 15, 90, 350, 1050, 2646, 5880, ...; 0, 1, 31, 301, 1701, 6951, 22827, 63987, ...; 0, 1, 63, 966, 7770, 42525, 179487, 627396, ...; ... MATHEMATICA Flatten[{1, Table[Sum[Binomial[n-1, k-1] * StirlingS2[n+k, k], {k, 1, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 11 2014 *) PROG (PARI) {a(n)=polcoeff(sum(k=0, n+1, (k^2*x)^k/(1-k*x)^k*exp(-k^2*x/(1-k*x+x*O(x^n)))/k!), n)} for(n=0, 25, print1(a(n), ", ")) (PARI) {Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!} {a(n)=if(n==0, 1, sum(k=1, n, Stirling2(n+k, k) * binomial(n-1, k-1)))} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A134055, A174845, A218667, A048993 (Stirling2). Sequence in context: A299077 A298997 A299664 * A365055 A196651 A197456 Adjacent sequences: A243939 A243940 A243941 * A243943 A243944 A243945 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 09 2014 STATUS approved

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Last modified August 13 05:07 EDT 2024. Contains 375113 sequences. (Running on oeis4.)