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Binomial transform of the partition numbers (A000041).
25

%I #35 Nov 02 2023 06:53:41

%S 1,2,5,13,34,88,225,569,1425,3538,8717,21331,51879,125474,301929,

%T 723144,1724532,4096210,9693455,22859524,53733252,125919189,294232580,

%U 685661202,1593719407,3695348909,8548564856,19732115915,45450793102,104481137953,239718272765

%N Binomial transform of the partition numbers (A000041).

%C Partial sums of A218482.

%C From _Vaclav Kotesovec_, Nov 02 2023: (Start)

%C Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then

%C Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where

%C g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).

%C Special cases:

%C p < 1/2, g(n) = 0

%C p = 1/2, g(n) = r^2/16

%C p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81

%C 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 p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))

%C 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 (End)

%H Vaclav Kotesovec, <a href="/A218481/b218481.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/(1-x)*Product_{n>=1} (1-x)^n / ((1-x)^n - x^n).

%F 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 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 G.f.: 1/(1-x)*exp( Sum_{n>=1} x^n/((1-x)^n - x^n) / n ).

%F 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 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 Logarithmic derivative yields A222115.

%F a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-1) / (n*sqrt(3)). - _Vaclav Kotesovec_, Jun 25 2015

%e 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 The g.f. equals the product:

%e 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 and also equals the series:

%e 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 The terms begin:

%e a(0) = 1*1,

%e a(1) = 1*1 + 1*1 = 2;

%e a(2) = 1*1 + 2*1 + 1*2 = 5;

%e a(3) = 1*1 + 3*1 + 3*2 + 1*3 = 13;

%e a(4) = 1*1 + 4*1 + 6*2 + 4*3 + 1*5 = 34; ...

%t Table[Sum[Binomial[n,k]*PartitionsP[k],{k,0,n}],{n,0,30}] (* _Vaclav Kotesovec_, Jun 25 2015 *)

%t 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 (PARI) {a(n)=sum(k=0,n,binomial(n,k)*numbpart(k))}

%o for(n=0,40,print1(a(n),", "))

%o (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 (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 (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 (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 (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 (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 Cf. A218482, A222115, A000041, A000203, A266497.

%Y Cf. A294466, A281425, A095051.

%Y Cf. A266232, A294467, A293467, A294468, A294500.

%K nonn,nice

%O 0,2

%A _Paul D. Hanna_, Oct 29 2012