A376095
a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1)^2 * a(k) * a(n-k-1).
Original entry on oeis.org
1, 1, 5, 54, 983, 26863, 1029188, 52747686, 3491367091, 290276997159, 29639219057133, 3648073361410412, 532858993269296500, 91147584892512564076, 18051321652239427195456, 4098339933686479506696526, 1057506667415381878759070811, 307764793378228160791205354175
Offset: 0
-
a[0] = 1; a[n_] := a[n] = Sum[(k + 1)^2 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 3 x^2 A[x] A'[x] + x^3 A[x] A''[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
A376097
a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1)^4 * a(k) * a(n-k-1).
Original entry on oeis.org
1, 1, 17, 1410, 364019, 228282823, 296324235500, 712075198644414, 2918094100584013255, 19151474626728425949663, 191553141880332262049655201, 2804913258838830873001491036584, 58168297154586087400230338311689652, 1661461159115675581245556180230933084340
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[(k + 1)^4 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
nmax = 13; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 15 x^2 A[x] A'[x] + 25 x^3 A[x] A''[x] + 10 x^4 A[x] A'''[x] + x^5 A[x] A''''[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
A376127
a(n) = 1 + Sum_{k=0..n-1} (k+1)^3 * a(k) * a(n-k-1).
Original entry on oeis.org
1, 2, 19, 565, 38056, 4886164, 1071397370, 370880032881, 191040201050842, 139853547948358801, 140279102716474353325, 187136598610376840549341, 323937672908434382002891895, 712668454800648677607151322833, 1957709831409075714559805601326566, 6613164804688226108094777888275765585
Offset: 0
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a[n_] := a[n] = 1 + Sum[(k + 1)^3 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
nmax = 15; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x] - 7 x^2 A'[x] - 6 x^3 A''[x] - x^4 A'''[x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
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