A351969
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * a(k) * a(n-4*k-1).
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 7, 22, 57, 128, 389, 1904, 9329, 38040, 132147, 542648, 3283633, 20997824, 114657097, 536178880, 2784500161, 19876061313, 153326461333, 1034551840125, 6051063487505, 38079448058864, 312420426286609, 2785055245002944, 22141255546319849
Offset: 0
-
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 4 k] a[k] a[n - 4 k - 1], {k, 0, Floor[(n - 1)/4]}]; Table[a[n], {n, 0, 28}]
A351968
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * a(k) * a(n-3*k-1).
Original entry on oeis.org
1, 1, 1, 1, 2, 6, 16, 37, 114, 478, 1907, 6777, 28414, 148580, 758930, 3580294, 18982050, 117888762, 720679726, 4193516446, 26798335830, 191775198574, 1353198262531, 9303932353127, 69303156652024, 559295471922890, 4454686099742810, 35198016469190740
Offset: 0
-
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 3 k] a[k] a[n - 3 k - 1], {k, 0, Floor[(n - 1)/3]}]; Table[a[n], {n, 0, 27}]
A352435
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * a(k) * a(n-2*k-1).
Original entry on oeis.org
1, 1, 2, 7, 32, 182, 1244, 9919, 90384, 926552, 10553728, 132231446, 1807390960, 26762801828, 426771821000, 7291604699407, 132885997278944, 2573145015936096, 52756125043795232, 1141727892772848248, 26009303834699461248, 622134297287753003008, 15589886235793001142016
Offset: 0
-
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 2 k + 1] a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 22}]
A138315
E.g.f. satisfies: A(x) = exp( Sum_{n>=0} a(n)*x^(2n+1)/(2n+1) ).
Original entry on oeis.org
1, 1, 1, 3, 9, 45, 225, 3015, 22545, 473625, 4637025, 191508075, 2288273625, 123786785925, 1735168572225, 276775700288175, 4433024011291425, 510439906229029425, 9196849128341801025, 3123191121957643317075
Offset: 0
E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 45*x^5/5! + 225*x^6/6! +...
Log(A(x)) = x + x^3/3 + x^5/5 + 3*x^7/7 + 9*x^9/9 + 45*x^11/11 + 225*x^13/13 +...
-
{a(n)=local(A=if(n==0,x,sum(k=0,n\2,a(k)*x^(2*k+1)/(2*k+1)))); n!*polcoeff(exp(A+x*O(x^n)),n)}
A351941
a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(k) * a(n-2*k-1).
Original entry on oeis.org
1, -1, 1, 0, -3, 8, -3, -57, 225, -96, -2991, 13873, -255, -313506, 1495089, 1473024, -54621231, 243365688, 705129201, -14109279483, 53228648865, 349791931434, -5000315242479, 13572033641204, 204954070915977, -2294997521498172, 2691551249257017
Offset: 0
-
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n - 1, 2 k] a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 26}]
Showing 1-5 of 5 results.