A178325
G.f.: A(x) = Sum_{n>=0} x^n/(1-x)^(n^2).
Original entry on oeis.org
1, 1, 2, 6, 21, 83, 363, 1730, 8889, 48829, 284858, 1755325, 11374092, 77208275, 547261631, 4039201624, 30967330941, 246084049137, 2023030659970, 17175765057532, 150367445873108, 1355528352031358, 12566899017130088
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 83*x^5 + 363*x^6 +...
A(x) = 1 + (x-x^2)*((1-x)-x)/((1-x)^3-x) + (x-x^2)^2*((1-x)-x)*((1-x)^5-x)/(((1-x)^3-x)*((1-x)^7-x)) + (x-x^2)^3*((1-x)-x)*((1-x)^5-x)*((1-x)^9-x)/(((1-x)^3-x)*((1-x)^7-x)*((1-x)^11-x)) +...
-
nn=22;CoefficientList[Series[Sum[x^k/(1-x)^(k^2),{k,0,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Oct 09 2013 *)
-
{a(n)=sum(k=0,n,binomial((n-k)^2+k-1,k))}
-
{a(n)=polcoeff(sum(m=0,n,x^m/(1-x+x*O(x^n))^(m^2)),n)}
-
{a(n)=polcoeff(sum(m=0,n,(x-x^2)^m*prod(k=1,m,((1-x)^(4*k-3)-x)/((1-x)^(4*k-1)-x +x*O(x^n)))),n)}
A227934
G.f.: Sum_{n>=0} x^n / (1-x)^(n^4).
Original entry on oeis.org
1, 1, 2, 18, 219, 4395, 129280, 4970984, 257765641, 16781325293, 1348125117404, 132465548869248, 15490711962965785, 2134540479514352751, 343307151209151099650, 63606662918084631874716, 13470938654397531939066909, 3238387688528230753569245297, 876825599524773154743990986391
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 18*x^3 + 219*x^4 + 4395*x^5 + 129280*x^6 +...
where
A(x) = 1 + x/(1-x) + x^2/(1-x)^16 + x^3/(1-x)^81 + x^4/(1-x)^256 + x^5/(1-x)^625 + x^6/(1-x)^1296 + x^7/(1-x)^2401 +...
-
{a(n)=polcoeff(sum(k=0,n,x^k/(1-x+x*O(x^n))^(k^4)),n)}
for(n=0,20,print1(a(n),", "))
-
{a(n)=sum(k=0,n,binomial(k^4+n-k-1, n-k))}
for(n=0,20,print1(a(n),", "))
A227935
G.f.: Sum_{n>=0} x^n / (1-x)^(n^5).
Original entry on oeis.org
1, 1, 2, 34, 773, 36656, 3001377, 333647780, 58561139773, 13838291852092, 4280413527001849, 1779704699369214238, 931039792575220097699, 604786686422678514970170, 489307443863919174036440087, 478922652139578822529676247092, 560120417434857039499787289137249
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 34*x^3 + 773*x^4 + 36656*x^5 + 3001377*x^6 +...
where
A(x) = 1 + x/(1-x) + x^2/(1-x)^32 + x^3/(1-x)^243 + x^4/(1-x)^1024 + x^5/(1-x)^3125 + x^6/(1-x)^7776 +...
-
{a(n)=polcoeff(sum(k=0,n,x^k/(1-x+x*O(x^n))^(k^5)),n)}
for(n=0,20,print1(a(n),", "))
-
{a(n)=sum(k=0,n,binomial(k^5+n-k-1, n-k))}
for(n=0,20,print1(a(n),", "))
A230049
Triangle such that the g.f. of column k equals 1/(1-x)^(k^3) for k>=0, as read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 1, 8, 1, 0, 1, 36, 27, 1, 0, 1, 120, 378, 64, 1, 0, 1, 330, 3654, 2080, 125, 1, 0, 1, 792, 27405, 45760, 7875, 216, 1, 0, 1, 1716, 169911, 766480, 333375, 23436, 343, 1, 0, 1, 3432, 906192, 10424128, 10668000, 1703016, 58996, 512, 1, 0, 1, 6435, 4272048, 119877472, 275234400, 93240126, 6784540, 131328, 729, 1
Offset: 0
Triangle begins:
1;
0, 1;
0, 1, 1;
0, 1, 8, 1;
0, 1, 36, 27, 1;
0, 1, 120, 378, 64, 1;
0, 1, 330, 3654, 2080, 125, 1;
0, 1, 792, 27405, 45760, 7875, 216, 1;
0, 1, 1716, 169911, 766480, 333375, 23436, 343, 1;
0, 1, 3432, 906192, 10424128, 10668000, 1703016, 58996, 512, 1;
0, 1, 6435, 4272048, 119877472, 275234400, 93240126, 6784540, 131328, 729, 1; ...
-
{T(n, k) = polcoeff(1/(1-x+x*O(x^n))^(k^3), n-k)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
-
{T(n, k) = binomial(k^3+n-k-1, n-k)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
Showing 1-4 of 4 results.
Comments