A172385
a(n) = 1 if n=0, otherwise Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k) *(-1)^k*a(n-1-2k).
Original entry on oeis.org
1, 1, 1, 0, -2, -4, -1, 14, 34, 2, -189, -439, 263, 3796, 6997, -14437, -96643, -106774, 671097, 2800836, 57519, -31088662, -82674287, 155322877, 1455331563, 1936970102, -14267868745, -66446614533, 19215003803, 1037638182571, 2654391633166, -8675836955120, -67833031653088
Offset: 0
Eigensequence for the number triangle
1;
1, 0;
0, 1, 0;
-1, 0, 1, 0;
0, -2, 0, 1, 0;
1, 0, -3, 0, 1, 0;
0, 3, 0, -4, 0, 1, 0;
-1, 0, 6, 0, -5, 0, 1, 0;
0, -4, 0, 10, 0, -6, 0, 1, 0;
1, 0, -10, 0, 15, 0, -7, 0, 1, 0;
0, 5, 0, -20, 0, 21, 0, -8, 0, 1, 0;
(augmented version of Riordan array (1/(1+x^2), x/(1+x^2)).
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a[n_] := If[n == 0, 1, Sum[Binomial[n - k - 1, k]*(-1)^k*a[n - 2*k - 1], {k, 0, Floor[(n - 1)/2]}]]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Oct 07 2018 *)
A360890
G.f. satisfies A(x) = 1 + x/(1 - x^3) * A(x/(1 - x^3)).
Original entry on oeis.org
1, 1, 1, 1, 2, 4, 7, 12, 25, 55, 115, 245, 564, 1331, 3103, 7407, 18388, 46198, 116503, 299966, 789426, 2095941, 5616114, 15299205, 42255533, 117689096, 331204936, 944052610, 2718150015, 7891518587, 23137661717, 68524545717, 204645635263, 616098009473
Offset: 0
-
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-1-2*j, j)*v[i-3*j])); v;
A352864
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k,k) * a(n-2*k-1).
Original entry on oeis.org
1, 1, 1, 3, 6, 13, 34, 84, 230, 653, 1893, 5794, 18080, 58345, 193761, 657959, 2295398, 8177305, 29775086, 110676222, 419169483, 1617868052, 6353518921, 25376986471, 103017630200, 424704411564, 1777458163195, 7546547411488, 32490058003914, 141774055915497, 626739661952337
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - k, k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 30}]
nmax = 30; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x^2)]/(1 - x^2)^2 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
A360891
G.f. satisfies A(x) = 1 + x/(1 - x^4) * A(x/(1 - x^4)).
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 4, 7, 11, 17, 32, 66, 132, 247, 463, 937, 2001, 4248, 8758, 18166, 39181, 87096, 193493, 425468, 942610, 2137196, 4930702, 11393809, 26280211, 61089849, 144157779, 343855549, 822430473, 1970839418, 4757600242, 11605042346, 28516751351
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)\4, binomial(i-1-3*j, j)*v[i-4*j])); v;
Showing 1-4 of 4 results.