cp's OEIS Frontend

A218481 Binomial transform of the partition numbers (A000041).

%I A218481 #35 Nov 02 2023 06:53:41
%S A218481 1,2,5,13,34,88,225,569,1425,3538,8717,21331,51879,125474,301929,
%T A218481 723144,1724532,4096210,9693455,22859524,53733252,125919189,294232580,
%U A218481 685661202,1593719407,3695348909,8548564856,19732115915,45450793102,104481137953,239718272765
%N A218481 Binomial transform of the partition numbers (A000041).
%C A218481 Partial sums of A218482.
%C A218481 From _Vaclav Kotesovec_, Nov 02 2023: (Start)
%C A218481 Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then
%C A218481 Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where
%C A218481 g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).
%C A218481 Special cases:
%C A218481 p < 1/2, g(n) = 0
%C A218481 p = 1/2, g(n) = r^2/16
%C A218481 p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81
%C A218481 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
%C A218481 p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))
%C A218481 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
%C A218481 (End)
%H A218481 Vaclav Kotesovec, <a href="/A218481/b218481.txt">Table of n, a(n) for n = 0..1000</a>
%F A218481 G.f.: 1/(1-x)*Product_{n>=1} (1-x)^n / ((1-x)^n - x^n).
%F A218481 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).
%F A218481 G.f.: 1/(1-x)*Sum_{n>=0} x^(n^2) * (1-x)^n / Product_{k=1..n} ((1-x)^k - x^k)^2.
%F A218481 G.f.: 1/(1-x)*exp( Sum_{n>=1} x^n/((1-x)^n - x^n) / n ).
%F A218481 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).
%F A218481 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.
%F A218481 Logarithmic derivative yields A222115.
%F A218481 a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-1) / (n*sqrt(3)). - _Vaclav Kotesovec_, Jun 25 2015
%e A218481 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 +...
%e A218481 The g.f. equals the product:
%e A218481 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) *...
%e A218481 and also equals the series:
%e A218481 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 +...).
%e A218481 The terms begin:
%e A218481 a(0) = 1*1,
%e A218481 a(1) = 1*1 + 1*1 = 2;
%e A218481 a(2) = 1*1 + 2*1 + 1*2 = 5;
%e A218481 a(3) = 1*1 + 3*1 + 3*2 + 1*3 = 13;
%e A218481 a(4) = 1*1 + 4*1 + 6*2 + 4*3 + 1*5 = 34; ...
%t A218481 Table[Sum[Binomial[n,k]*PartitionsP[k],{k,0,n}],{n,0,30}] (* _Vaclav Kotesovec_, Jun 25 2015 *)
%t A218481 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 *)
%o A218481 (PARI) {a(n)=sum(k=0,n,binomial(n,k)*numbpart(k))}
%o A218481 for(n=0,40,print1(a(n),", "))
%o A218481 (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)}
%o A218481 (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)}
%o A218481 (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)}
%o A218481 (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)}
%o A218481 (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)}
%o A218481 (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)}
%Y A218481 Cf. A218482, A222115, A000041, A000203, A266497.
%Y A218481 Cf. A294466, A281425, A095051.
%Y A218481 Cf. A266232, A294467, A293467, A294468, A294500.
%K A218481 nonn,nice
%O A218481 0,2
%A A218481 _Paul D. Hanna_, Oct 29 2012