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 A218481 Binomial transform of the partition numbers (A000041). 25
 1, 2, 5, 13, 34, 88, 225, 569, 1425, 3538, 8717, 21331, 51879, 125474, 301929, 723144, 1724532, 4096210, 9693455, 22859524, 53733252, 125919189, 294232580, 685661202, 1593719407, 3695348909, 8548564856, 19732115915, 45450793102, 104481137953, 239718272765 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Partial sums of A218482. From Vaclav Kotesovec, Nov 02 2023: (Start) Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)). Special cases: p < 1/2, g(n) = 0 p = 1/2, g(n) = r^2/16 p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81 p = 3/4, g(n) = 9*r^2*sqrt(n)/(64*sqrt(2)) - 27*r^3*n^(1/4)/(2048*2^(1/4)) + 81*r^4/65536 p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5)) p = 4/5, g(n) = 2^(7/5)*r^2*n^(3/5)/25 - 4*2^(3/5)*r^3*n^(2/5)/625 + 8*2^(4/5)*r^4*n^(1/5)/15625 - 32*r^5/390625 (End) LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 FORMULA G.f.: 1/(1-x)*Product_{n>=1} (1-x)^n / ((1-x)^n - x^n). G.f.: 1/(1-x)*Sum_{n>=0} x^n * (1-x)^(n*(n-1)/2) / Product_{k=1..n} ((1-x)^k - x^k). G.f.: 1/(1-x)*Sum_{n>=0} x^(n^2) * (1-x)^n / Product_{k=1..n} ((1-x)^k - x^k)^2. G.f.: 1/(1-x)*exp( Sum_{n>=1} x^n/((1-x)^n - x^n) / n ). G.f.: 1/(1-x)*exp( Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n ), where sigma(n) is the sum of divisors of n (A000203). G.f.: 1/(1-x)*Product_{n>=1} (1 + x^n/(1-x)^n)^A001511(n), where 2^A001511(n) is the highest power of 2 that divides 2*n. Logarithmic derivative yields A222115. a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-1) / (n*sqrt(3)). - Vaclav Kotesovec, Jun 25 2015 EXAMPLE G.f.: A(x) = 1 + 2*x + 5*x^2 + 13*x^3 + 34*x^4 + 88*x^5 + 225*x^6 + 569*x^7 +... The g.f. equals the product: A(x) = 1/((1-x)-x) * (1-x)^2/((1-x)^2-x^2) * (1-x)^3/((1-x)^3-x^3) * (1-x)^4/((1-x)^4-x^4) * (1-x)^5/((1-x)^5-x^5) * (1-x)^6/((1-x)^6-x^6) * (1-x)^7/((1-x)^7-x^7) *... and also equals the series: A(x) = 1/(1-x) * (1 + x*(1-x)/((1-x)-x)^2 + x^4*(1-x)^2/(((1-x)-x)*((1-x)^2-x^2))^2 + x^9*(1-x)^3/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3))^2 + x^16*(1-x)^4/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3)*((1-x)^4-x^4))^2 +...). The terms begin: a(0) = 1*1, a(1) = 1*1 + 1*1 = 2; a(2) = 1*1 + 2*1 + 1*2 = 5; a(3) = 1*1 + 3*1 + 3*2 + 1*3 = 13; a(4) = 1*1 + 4*1 + 6*2 + 4*3 + 1*5 = 34; ... MATHEMATICA Table[Sum[Binomial[n, k]*PartitionsP[k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jun 25 2015 *) nmax = 30; CoefficientList[Series[Sum[PartitionsP[k] * x^k / (1-x)^(k+1), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2022 *) PROG (PARI) {a(n)=sum(k=0, n, binomial(n, k)*numbpart(k))} for(n=0, 40, print1(a(n), ", ")) (PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(1/(1-X)*prod(k=1, n, (1-x)^k/((1-x)^k-X^k)), n)} (PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(1/(1-X)*sum(m=0, n, x^m*(1-x)^(m*(m-1)/2)/prod(k=1, m, ((1-x)^k - X^k))), n)} (PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(1/(1-X)*sum(m=0, n, x^(m^2)*(1-X)^m/prod(k=1, m, ((1-x)^k - x^k)^2)), n)} (PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(1/(1-X)*exp(sum(m=1, n+1, x^m/((1-x)^m-X^m)/m)), n)} (PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(1/(1-X)*exp(sum(m=1, n+1, sigma(m)*x^m/(1-X)^m/m)), n)} (PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(1/(1-X)*prod(k=1, n, (1 + x^k/(1-X)^k)^valuation(2*k, 2)), n)} CROSSREFS Cf. A218482, A222115, A000041, A000203, A266497. Cf. A294466, A281425, A095051. Cf. A266232, A294467, A293467, A294468, A294500. Sequence in context: A318234 A371426 A027931 * A267905 A209230 A103142 Adjacent sequences: A218478 A218479 A218480 * A218482 A218483 A218484 KEYWORD nonn,nice AUTHOR Paul D. Hanna, Oct 29 2012 STATUS approved

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Last modified September 11 02:28 EDT 2024. Contains 375813 sequences. (Running on oeis4.)