A298695
G.f.: Sum_{n>=0} binomial(n^2, n) * x^n / (1 + x)^(n^2).
Original entry on oeis.org
1, 1, 5, 61, 1123, 27671, 853411, 31603447, 1365807689, 67469763889, 3749935785301, 231591200859701, 15733654527061483, 1166102347943957815, 93629607937879486019, 8096167402408961507311, 750088483178476669111441, 74127049788588758257392161, 7783440821906363883725443813, 865349148215025766722403077229, 101553078711812924877087765912371
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 61*x^3 + 1123*x^4 + 27671*x^5 + 853411*x^6 + 31603447*x^7 + 1365807689*x^8 + 67469763889*x^9 + 3749935785301*x^10 + ...
such that
A(x) = 1 + C(1,1)*x/(1+x) + C(4,2)*x^2/(1+x)^4 + C(9,3)*x^3/(1+x)^9 + C(16,4)*x^4/(1+x)^16 + C(25,5)*x^5/(1+x)^25 + C(36,6)*x^6/(1+x)^36 + ...
more explicitly,
A(x) = 1 + x/(1+x) + 6*x^2/(1+x)^4 + 84*x^3/(1+x)^9 + 1820*x^4/(1+x)^16 + 53130*x^5/(1+x)^25 + 1947792*x^6/(1+x)^36 + ... + A014062(n)*x^n/(1+x)^(n^2) + ...
-
terms = 21; s = Sum[Binomial[n^2, n]*x^n/(1 + x)^(n^2), {n, 0, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Feb 06 2018 *)
-
{a(n) = my(A = sum(m=0,n,binomial(m^2,m)*x^m/(1+x +x*O(x^n))^(m^2) ) ); polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))
A299435
G.f.: Sum_{n>=0} binomial((n+1)^2, n)/(n+1) * x^n / (1 + x)^((n+1)^2).
Original entry on oeis.org
1, 1, 5, 51, 791, 16711, 449575, 14738537, 570860449, 25534320961, 1296145448621, 73644069770107, 4631766294581959, 319523289664700279, 23992478864877747151, 1948216141720780468561, 170121586262631029818433, 15897659114382366967974145, 1583109774987253349677203349, 167363833662976153803805436291, 18721216520653602533835176495671
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 51*x^3 + 791*x^4 + 16711*x^5 + 449575*x^6 + 14738537*x^7 + 570860449*x^8 + 25534320961*x^9 + 1296145448621*x^10 + ...
such that
A(x) = 1/(1+x) + C(4,1)/2*x/(1+x)^4 + C(9,2)/3*x^2/(1+x)^9 + C(16,3)/4*x^3/(1+x)^16 + C(25,4)/5*x^4/(1+x)^25 + C(36,5)/6*x^5/(1+x)^36 + C(49,6)/7*x^6/(1+x)^49 + ...
more explicitly,
A(x) = 1/(1+x) + 2*x/(1+x)^4 + 12*x^2/(1+x)^9 + 140*x^3/(1+x)^16 + 2530*x^4/(1+x)^25 + 62832*x^5/(1+x)^36 + 1997688*x^6/(1+x)^49 + ... + A091144(n+1)*x^n/(1+x)^((n+1)^2) + ...
-
{a(n) = my(A = sum(m=0,n,binomial((m+1)^2,m)/(m+1)*x^m/(1+x +x*O(x^n))^((m+1)^2) ) ); polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))
A352700
G.f.: Sum_{n>=0} binomial((n+1)*(2*n+1),n)/(2*n+1) * x^n / C(x)^(n*(2*n+1)+1), where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).
Original entry on oeis.org
1, 1, 12, 239, 7178, 296092, 15666162, 1014796995, 77899495174, 6919858148750, 698584345392968, 79022119891573410, 9902447587480555624, 1361894352334815968554, 203969111022547680433454, 33047362680815865252524643, 5759708920548423261284008230
Offset: 0
G.f.: A(x) = 1 + x + 12*x^2 + 239*x^3 + 7178*x^4 + 296092*x^5 + 15666162*x^6 + 1014796995*x^7 + 77899495174*x^8 + 6919858148750*x^9 + ...
where
A(x) = 1/C(x) + 2*x/C(x)^4 + 21*x^2/C(x)^11 + 468*x^3/C(x)^22 + 16555*x^4/C(x)^37 + 812448*x^5/C(x)^56 + 51274146*x^6/C(x)^79 + 3965837928*x^7/C(x)^106 + ... + (n+1)*A299429(n)*x^n/C(x)^(n*(2*n+1)+1) + ...
and
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + ... + A000108(n)*x^n + ...
Congruence modulo 3.
(1) It appears that A(x)^3 is congruent to A(x^3) modulo 3, where
A(x)^3 = 1 + 3*x + 39*x^2 + 790*x^3 + 23436*x^4 + 949701*x^5 + 49503687*x^6 + 3171679536*x^7 + 241578165750*x^8 + 21340270771814*x^9 + ...
and
(A(x)^3 - A(x^3))/3 = x + 13*x^2 + 263*x^3 + 7812*x^4 + 316567*x^5 + 16501225*x^6 + 1057226512*x^7 + 80526055250*x^8 + 7113423590525*x^9 + ...
(2) Also, g.f. A(x) seems to be congruent to C(x) + x^2*C(x)^3, where
C(x) + x^2*C(x)^3 = 1 + x + 3*x^2 + 8*x^3 + 23*x^4 + 70*x^5 + 222*x^6 + 726*x^7 + 2431*x^8 + 8294*x^9 + ... + (C(2*n,n)/(n+1) + C(2*n-1,n-2)*3/(2*n-1))*x^n + ...
-
{a(n) = my(C = (1 - sqrt(1-4*x +O(x^(n+3))))/(2*x),
A = sum(m=0,n, binomial((m+1)*(2*m+1),m)/(2*m+1) * x^m/C^(m*(2*m+1)+1))); polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
Showing 1-3 of 3 results.
Comments