A351028
G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 2*x)) / (1 - 2*x).
Original entry on oeis.org
0, 1, 0, 1, 4, 13, 44, 173, 792, 4009, 21608, 122761, 737340, 4696341, 31665076, 224846037, 1672266352, 12976252561, 104816144656, 880061135057, 7670326372532, 69286959112797, 647568753568636, 6251768635591613, 62255057942504968, 638658964709824185
Offset: 0
-
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, 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] = x + x^2 A[x/(1 - 2 x)]/(1 - 2 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] 2^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}];
(* another pprogram *)
B[x_] := BesselK[0, 1]*BesselI[0, Exp[x]] - BesselI[0, 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 *)
A351053
G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 3*x)) / (1 - 3*x).
Original entry on oeis.org
0, 1, 0, 1, 6, 28, 126, 613, 3438, 22159, 157362, 1189126, 9436320, 78690781, 692478684, 6439539457, 63106488618, 648453907216, 6952719052134, 77521908188737, 897132401326458, 10764085132255807, 133774484448519294, 1720018195807299418, 22847325911461934352
Offset: 0
-
nmax = 24; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 3 x)]/(1 - 3 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] 3^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 24}]
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
-
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}]
A351132
G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).
Original entry on oeis.org
0, 1, 0, 1, 10, 76, 530, 3701, 27810, 237151, 2316350, 25135126, 292106400, 3559029501, 45211131460, 600619791201, 8384107777030, 123237338584576, 1904128564485610, 30789744821412401, 518479182191232950, 9057086806410632751, 163745788914416588050
Offset: 0
-
nmax = 22; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 5 x)]/(1 - 5 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] 5^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]
Showing 1-4 of 4 results.
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