A100095 -id:A100095 - cp's OEIS frontend

A100096 An inverse Chebyshev transform of the Jacobsthal numbers.

0, 1, 1, 6, 9, 36, 66, 218, 449, 1332, 2946, 8196, 18954, 50688, 120576, 314586, 761889, 1957092, 4794426, 12194828, 30093854, 76067256, 188595276, 474810276, 1180734234, 2965094536, 7387570516, 18521858088, 46203981924, 115721310552

Offset: 0

Paul Barry, Nov 03 2004

Image of x/(1-x-2*x^2) under the transform g(x)->(1/sqrt(1-4*x^2))*g(x*c(x^2)), where c(x) is the g.f. of the Catalan numbers A000108. This is the inverse of the Chebyshev transform which takes A(x) to ((1-x^2)/(1+x^2))*A(x/(1+x^2)).

Hankel transform is (-1)^n*n. Hankel transform of a(n+1) is A141124. - Paul Barry, Jun 05 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A100095, A100097.

CoefficientList[Series[x*Sqrt[1-4*x^2]*(3*Sqrt[1-4*x^2]+2*x+1)/(2*(4*x^2-1)*(10*x^2+x-2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

G.f.: x*sqrt(1-4*x^2)*(3*sqrt(1-4*x^2)+2*x+1)/(2*(4*x^2-1)*(10*x^2+x-2)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*A001045(n-2*k).
Conjecture: 2*(-n+1)*a(n) +(n+3)*a(n-1) +18*(n-2)*a(n-2) +4*(-n-2)*a(n-3) +40*(-n+3)*a(n-4)=0. - R. J. Mathar, Nov 24 2012
a(n) ~ (5/2)^n / 3. - Vaclav Kotesovec, Feb 12 2014

A100097 An inverse Chebyshev transform of the Pell numbers.

Original entry on oeis.org

0, 1, 2, 8, 20, 64, 172, 512, 1416, 4096, 11468, 32768, 92248, 262144, 739832, 2097152, 5925520, 16777216, 47429900, 134217728, 379536440, 1073741824, 3036661032, 8589934592, 24294699120, 68719476736, 194363001272, 549755813888, 1554924811376, 4398046511104

Offset: 0

Paul Barry, Nov 03 2004

Image of x/(1-2*x-x^2) under the transform g(x)->(1/sqrt(1-4*x^2))*g(x*c(x^2)), where c(x) is the g.f. of the Catalan numbers A000108. This is the inverse of the Chebyshev transform which takes A(x) to ((1-x^2)/(1+x^2))*A(x/(1+x^2)).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A100095, A100096.

CoefficientList[Series[x*Sqrt[1-4*x^2]*(Sqrt[1-4*x^2]+2*x)/((1-4*x^2)*(1-8*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

G.f.: x*sqrt(1-4*x^2)*(sqrt(1-4*x^2)+2*x)/((1-4*x^2)*(1-8*x^2)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*A000129(n-2*k).
Conjecture: (-n+2)*a(n) +(-n+3)*a(n-1) +4*(3*n-7)*a(n-2) +4*(3*n-10)*a(n-3) +32*(-n+3)*a(n-4) +32*(-n+4)*a(n-5)=0. - R. J. Mathar, Nov 24 2012
Recurrence: (n-2)*a(n) = 4*(3*n-7)*a(n-2) - 32*(n-3)*a(n-4). - Vaclav Kotesovec, Feb 12 2014
a(n) ~ 2^((3*n-3)/2). - Vaclav Kotesovec, Feb 12 2014
a(2*n) = 8^n/(2*sqrt(2)) - 2^n * (2*n-1)!! * hypergeom([1, n+1/2], [n+1], 1/2)/(4*n!), a(2*n+1) = 8^n. - Vladimir Reshetnikov, Oct 13 2016

A373583 Expansion of 1 / ( (1 - 16x^4) (1 - x/(1 - 16*x^4)^(1/4)) ).

Original entry on oeis.org

1, 1, 1, 1, 17, 21, 25, 29, 289, 397, 521, 661, 4913, 7229, 10137, 13701, 83521, 129133, 190249, 269877, 1419857, 2280125, 3492281, 5149701, 24137569, 39950221, 63153481, 96159061, 410338673, 696126557, 1129839065, 1767607973, 6975757441, 12080257069

Offset: 0

Seiichi Manyama, Jun 11 2024

nonn

Cf. A000079, A100095, A373278, A373621.

Cf. A373627.

a(n) = sum(k=0, n\4, 16^k*binomial(n/4, k));

a(4*n) = 17^n for n >= 0.
a(n) = Sum_{k=0..floor(n/4)} 16^k * binomial(n/4,k).
a(n) == 1 (mod 4).

A141125 Hankel transform of a transform of Fibonacci numbers.

Original entry on oeis.org

1, 4, -4, -16, 16, 64, -64, -256, 256, 1024, -1024, -4096, 4096, 16384, -16384, -65536, 65536, 262144, -262144, -1048576, 1048576, 4194304, -4194304, -16777216, 16777216, 67108864, -67108864, -268435456, 268435456, 1073741824, -1073741824, -4294967296, 4294967296

Offset: 0

Paul Barry, Jun 05 2008

Hankel transform of A100095(n+1).

Index entries for linear recurrences with constant coefficients, signature (0,-4).

Cf. A100095, A141124.

LinearRecurrence[{0,-4},{1,4},33] (* or *) CoefficientList[Series[(1+4x)/(1+4x^2),{x,0,32}],x] (* James C. McMahon, Jul 17 2025 *)

G.f.: (1+4x)/(1+4x^2).
a(n) = 2^n*(cos(Pi*n/2)+2*sin(Pi*n/2)).
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
E.g.f.: cos(2*x) + 2*sin(2*x). - Stefano Spezia, Jul 18 2025

More terms from James C. McMahon, Jul 18 2025

A373278 Expansion of 1 / ( (1 - 9x^3) (1 - x/(1 - 9*x^3)^(1/3)) ).

Original entry on oeis.org

1, 1, 1, 10, 13, 16, 100, 148, 205, 1000, 1606, 2410, 10000, 17005, 27070, 100000, 177421, 295648, 1000000, 1833178, 3168538, 10000000, 18811948, 33503020, 100000000, 192080866, 350707345, 1000000000, 1953820210, 3642942040, 10000000000, 19815499120, 37611477133

Offset: 0

Seiichi Manyama, Jun 11 2024

nonn

Cf. A000079, A100095, A373583, A373621.

Cf. A011557, A371458.

a(n) = sum(k=0, n\3, 9^k*binomial(n/3, k));

a(3*n) = 10^n for n >= 0.
a(n) = Sum_{k=0..floor(n/3)} 9^k * binomial(n/3,k).
a(n) == 1 (mod 3).
D-finite with recurrence (n-1)*(n-2)*a(n) +2*(-14*n^2+69*n-91)*a(n-3) +9*(n-3)*(29*n-114)*a(n-6) -810*(n-3)*(n-6)*a(n-9)=0. - R. J. Mathar, Jun 21 2024

A373621 Expansion of 1 / ( (1 - 25x^5) (1 - x/(1 - 25*x^5)^(1/5)) ).

Original entry on oeis.org

1, 1, 1, 1, 1, 26, 31, 36, 41, 46, 676, 881, 1111, 1366, 1646, 17576, 24281, 32386, 42016, 53296, 456976, 658806, 916411, 1238666, 1635071, 11881376, 17706456, 25462936, 35569066, 48496846, 308915776, 472880356, 698851961, 1003283216, 1405555496, 8031810176

Offset: 0

Seiichi Manyama, Jun 11 2024

nonn

Cf. A000079, A100095, A373278, A373583.

Cf. A373510.

a(n) = sum(k=0, n\5, 25^k*binomial(n/5, k));

a(5*n) = 26^n for n >= 0.
a(n) = Sum_{k=0..floor(n/5)} 25^k * binomial(n/5,k).
a(n) == 1 (mod 5).

cp's OEIS Frontend

A100096 An inverse Chebyshev transform of the Jacobsthal numbers.

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A100097 An inverse Chebyshev transform of the Pell numbers.

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A373583 Expansion of 1 / ( (1 - 16*x^4) * (1 - x/(1 - 16*x^4)^(1/4)) ).

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A141125 Hankel transform of a transform of Fibonacci numbers.

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A373278 Expansion of 1 / ( (1 - 9*x^3) * (1 - x/(1 - 9*x^3)^(1/3)) ).

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A373621 Expansion of 1 / ( (1 - 25*x^5) * (1 - x/(1 - 25*x^5)^(1/5)) ).

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Formula

A373583 Expansion of 1 / ( (1 - 16x^4) (1 - x/(1 - 16*x^4)^(1/4)) ).

A373278 Expansion of 1 / ( (1 - 9x^3) (1 - x/(1 - 9*x^3)^(1/3)) ).

A373621 Expansion of 1 / ( (1 - 25x^5) (1 - x/(1 - 25*x^5)^(1/5)) ).