A351143
G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 2*x)) / (1 - 2*x).
Original entry on oeis.org
1, 0, 1, 2, 5, 16, 61, 258, 1177, 5776, 30537, 173394, 1050045, 6732608, 45459493, 322141106, 2390075249, 18525967328, 149684238801, 1257802518754, 10969260208565, 99100423076912, 926030783479629, 8937741026924450, 88988433270106249, 912906193294355952
Offset: 0
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bintr:= proc(p) local b;
b:= proc(n) option remember; add(p(k)*binomial(n, k), k=0..n) end
end:
b:= (bintr@@2)(a):
a:= n-> `if`(n<2, 1-n, b(n-2)):
seq(a(n), n=0..25); # Alois P. Heinz, Apr 07 2025
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nmax = 25; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 2^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
(* another program *)
B[x_] := BesselK[1, 1]*BesselI[0, Exp[x]] + BesselI[1, 1]*BesselK[0, Exp[x]];
a[n_] := SeriesCoefficient[FullSimplify[Series[B[x], {x, 0, n}]], n] n!
Table[a[n], {n, 0, 30}] (* Ven Popov, Apr 25 2025 *)
A351144
G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 3*x)) / (1 - 3*x).
Original entry on oeis.org
1, 0, 1, 3, 10, 39, 181, 972, 5797, 37389, 258202, 1905681, 15016465, 125920872, 1117950913, 10452866439, 102485649754, 1050464300187, 11231883627301, 125055844922916, 1447371528438565, 17382103226123313, 216221862096537994, 2781342531957176085, 36942930754308211969
Offset: 0
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nmax = 24; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 3 x)]/(1 - 3 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 3^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 24}]
A351151
G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).
Original entry on oeis.org
1, 0, 1, 5, 26, 145, 901, 6420, 52501, 480955, 4795626, 51066375, 576182001, 6879462680, 86955722401, 1162559359745, 16392133866026, 242734091500445, 3758825675820501, 60660434188558780, 1017770666417312501, 17725289455315892375, 320047193447632729626
Offset: 0
-
nmax = 22; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 5 x)]/(1 - 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 5^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]
A351152
G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 6*x)) / (1 - 6*x).
Original entry on oeis.org
1, 0, 1, 6, 37, 240, 1693, 13446, 122329, 1261104, 14332681, 175123446, 2267871517, 30981705984, 446571784261, 6798161166486, 109220619908593, 1846729159654560, 32726973173941585, 605358657750562470, 11648701234354836565, 232655173657593759312
Offset: 0
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nmax = 21; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 6 x)]/(1 - 6 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 6^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 21}]
A351128
G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 4*x)) / (1 - 4*x).
Original entry on oeis.org
0, 1, 0, 1, 8, 49, 280, 1649, 10800, 81505, 696400, 6472033, 63562872, 652984977, 7026210728, 79547049681, 949709767904, 11936248012993, 157219119485216, 2159448120457409, 30811324011852136, 455635009201780977, 6975424580445456056, 110478282815356437809
Offset: 0
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nmax = 23; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 4 x)]/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 4^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 23}]
Showing 1-5 of 5 results.
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