A125274
Eigensequence of triangle A078812: a(n) = Sum_{k=0..n-1} A078812(n-1,k)*a(k) for n > 0 with a(0)=1.
Original entry on oeis.org
1, 1, 3, 10, 42, 210, 1199, 7670, 54224, 418744, 3499781, 31425207, 301324035, 3069644790, 33078375153, 375634524357, 4480492554993, 55971845014528, 730438139266281, 9935106417137098, 140553930403702487
Offset: 0
a(3) = 3*(1) + 4*(1) + 1*(3) = 10;
a(4) = 4*(1) + 10*(1) + 6*(3) + 1*(10) = 42;
a(5) = 5*(1) + 20*(1) + 21*(3) + 8*(10) + 1*(42) = 210.
Triangle A078812(n,k) = binomial(n+k+1, n-k) begins:
1;
2, 1;
3, 4, 1;
4, 10, 6, 1;
5, 20, 21, 8, 1;
6, 35, 56, 36, 10, 1; ...
where g.f. of column k = 1/(1-x)^(2*k+2).
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n+k, n-k-1] * a[k], {k, 0, n-1}]; Array[a, 20, 0] (* Amiram Eldar, Nov 24 2018 *)
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a(n)=if(n==0,1,sum(k=0,n-1, a(k)*binomial(n+k, n-k-1)))
A351817
G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)^4) / (1 - x)^4.
Original entry on oeis.org
1, 1, 5, 23, 139, 1052, 9166, 90073, 989205, 11981051, 158149438, 2255926638, 34549223880, 564898101239, 9812669832553, 180324597042263, 3492960489714519, 71092066388237562, 1516044005669227542, 33788707128788508476, 785270646437483414261, 18992014442689191510460
Offset: 0
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nmax = 21; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)^4]/(1 - x)^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 3 k + 2, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
A351818
G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)^5) / (1 - x)^5.
Original entry on oeis.org
1, 1, 6, 31, 211, 1841, 18547, 210664, 2682657, 37807531, 581985596, 9696297528, 173702897000, 3327063115248, 67790086866271, 1462900566163696, 33310115601839624, 797687851718024035, 20032231443590167914, 526189230537615409571, 14423255501358439152231
Offset: 0
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nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)^5]/(1 - x)^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 4 k + 3, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 20}]
A354695
G.f. A(x) satisfies: A(x) = 1 + x * A(x^3/(1 - x)^3) / (1 - x)^3.
Original entry on oeis.org
1, 1, 3, 6, 11, 21, 42, 87, 189, 432, 1018, 2415, 5694, 13297, 30768, 70626, 161011, 364977, 823536, 1851706, 4152972, 9298653, 20800758, 46516437, 104044590, 232856189, 521601174, 1169670645, 2626188319, 5904269526, 13292581605, 29968831278, 67663806228
Offset: 0
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, (i-1)\3, binomial(i+1, 3*j+2)*v[j+1])); v;
Showing 1-4 of 4 results.