A115962 -id:A115962 - cp's OEIS frontend

A128063 Hankel transform of A115962.

1, 2, 0, -8, -16, -32, -64, 0, 256, 512, 1024, 2048, 0, -8192, -16384, -32768, -65536, 0, 262144, 524288, 1048576, 2097152, 0, -8388608, -16777216, -33554432, -67108864, 0, 268435456, 536870912, 1073741824, 2147483648, 0, -8589934592, -17179869184

Offset: 0

Paul Barry, Feb 13 2007

Cf. A099443, A115962.

a(n) = 2^n*((1/2 - 3*sqrt(5)/10)*cos(3*Pi*n/5) + sqrt(1/10 - sqrt(5)/50)*sin(3*Pi*n/5) + (3*sqrt(5)/10 + 1/2)*cos(Pi*n/5) - sqrt(sqrt(5)/50 + 1/10)*sin(Pi*n/5));
a(n) = 2^n*Sum_{k=0..floor((n+2)/2)} binomial(n-k+2,k)*(-1)^k*Fibonacci(n-2k+3);
a(n) = 2^n*A099443(n+2).
Empirical g.f.: -(2*x-1)*(4*x^2 + 2*x + 1) / (16*x^4 - 8*x^3 + 4*x^2 - 2*x + 1). - Colin Barker, Jun 28 2013

A115951 Expansion of 1/sqrt(1-4xy-4x^2y).

Original entry on oeis.org

1, 0, 2, 0, 2, 6, 0, 0, 12, 20, 0, 0, 6, 60, 70, 0, 0, 0, 60, 280, 252, 0, 0, 0, 20, 420, 1260, 924, 0, 0, 0, 0, 280, 2520, 5544, 3432, 0, 0, 0, 0, 70, 2520, 13860, 24024, 12870, 0, 0, 0, 0, 0, 1260, 18480, 72072, 102960, 48620, 0, 0, 0, 0, 0, 252, 13860, 120120, 360360, 437580, 184756

Offset: 0

Paul Barry, Mar 14 2006

Row sums are A006139. Diagonal sums are A115962.

Coefficients of 2^n * P(n, x) with P the Legendre P polynomials. Reflection of triangle A008556. - Ralf Stephan, Apr 07 2016.

			Triangle begins
   1,
   0,  2,
   0,  2,  6,
   0,  0, 12,  20,
   0,  0,  6,  60,  70,
   0,  0,  0,  60, 280,  252,
   0,  0,  0,  20, 420, 1260, 924

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A006139 (row sums), A063007 (binomial transform), A115962 (diagonal sums).

/* As triangle */ [[Binomial(2*k,k)*Binomial(k,n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 03 2015

Table[Binomial[2k, k]Binomial[k, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* Michael De Vlieger, Sep 02 2015 *)

{T(n,k) = binomial(2*k,k)*binomial(k,n-k)}; \\ G. C. Greubel, May 06 2019

[[binomial(2*k,k)*binomial(k,n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 06 2019

Number triangle T(n,k) = C(2k,k)*C(k,n-k).
From Peter Bala, Sep 02 2015: (Start)
Binomial transform is A063007; equivalently, P * M = A063007, where P denotes Pascal's triangle A007318 and M denotes the present array.
P * M * P^-1 is a signed version of A063007. (End)

A377186 Expansion of 1/(1 - 4x^2 - 4x^3)^(3/2).

Original entry on oeis.org

1, 0, 6, 6, 30, 60, 170, 420, 1050, 2660, 6552, 16380, 40362, 99792, 245520, 603372, 1480050, 3624192, 8863712, 21647340, 52811616, 128700000, 313341756, 762206016, 1852565650, 4499346072, 10919990460, 26485897932, 64201490352, 155536089240, 376606931436

Offset: 0

Seiichi Manyama, Oct 19 2024

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1500

Cf. A115962, A377189, A377190.

Cf. A374497.

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^2 - 4*x^3)^(3/2))); // Vincenzo Librandi, May 11 2025

Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[k,n-2*k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, May 11 2025 *)

a(n) = sum(k=0, n\2, (2*k+1)*binomial(2*k, k)*binomial(k, n-2*k));

a(0) = 1, a(1) = 0, a(2) = 6; a(n) = (4*(n+1)*a(n-2) + 2*(2*n+3)*a(n-3))/n.

a(n) = Sum_{k=0..floor(n/2)} (2*k+1) * binomial(2*k,k) * binomial(k,n-2*k).

A361726 Diagonal of rational function 1/(1 - (1 + xy) (x^2 + y^2)).

Original entry on oeis.org

1, 0, 2, 4, 8, 24, 56, 144, 376, 960, 2512, 6560, 17184, 45248, 119296, 315392, 835552, 2217216, 5893568, 15687552, 41810944, 111567104, 298016512, 796832256, 2132456704, 5711486976, 15309014528, 41062927360, 110213725184, 295995574272, 795391639552

Offset: 0

Seiichi Manyama, Mar 22 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A137635, A361727.

Cf. A115962, A360266.

a(n) = sum(k=0, n\2, binomial(2*k, k)*binomial(2*k, n-2*k));

G.f.: 1/sqrt(1 - 4 * x^2 * (1+x)^2).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*k,k) * binomial(2*k,n-2*k).
a(n) ~ (1 + sqrt(3))^(n + 1/2) / (2*3^(1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 22 2023
n*a(n) = 2*(2*n-2)*a(n-2) + 4*(2*n-3)*a(n-3) + 2*(2*n-4)*a(n-4) for n > 3. - Seiichi Manyama, Mar 23 2023

A377189 Expansion of 1/(1 - 4x^2 - 4x^3)^(5/2).

Original entry on oeis.org

1, 0, 10, 10, 70, 140, 490, 1260, 3570, 9660, 25872, 69300, 182490, 480480, 1252680, 3255252, 8412690, 21655920, 55535480, 141921780, 361577216, 918529040, 2327337740, 5882631040, 14836032770, 37339221192, 93794645700, 235186913780, 588736957920, 1471462327160

Offset: 0

Seiichi Manyama, Oct 19 2024

nonn

Cf. A115962, A377186, A377190.

Cf. A374511.

a(n) = sum(k=0, n\2, (-4)^k*binomial(-5/2, k)*binomial(k, n-2*k));

a(0) = 1, a(1) = 0, a(2) = 10; a(n) = (4*(n+3)*a(n-2) + 2*(2*n+9)*a(n-3))/n.

a(n) = Sum_{k=0..floor(n/2)} (-4)^k * binomial(-5/2,k) * binomial(k,n-2*k).

A377190 Expansion of 1/(1 - 4x^2 - 4x^3)^(7/2).

Original entry on oeis.org

1, 0, 14, 14, 126, 252, 1050, 2772, 8778, 24948, 72072, 204204, 570570, 1585584, 4351776, 11879868, 32162130, 86582496, 231703472, 616900284, 1634721088, 4312944064, 11333823228, 29673291648, 77423101938, 201367680696, 522180220044, 1350350044316, 3482928560880

Offset: 0

Seiichi Manyama, Oct 19 2024

nonn

Cf. A115962, A377186, A377189.

Cf. A374513.

a(n) = sum(k=0, n\2, (-4)^k*binomial(-7/2, k)*binomial(k, n-2*k));

a(0) = 1, a(1) = 0, a(2) = 14; a(n) = (4*(n+5)*a(n-2) + 2*(2*n+15)*a(n-3))/n.

a(n) = Sum_{k=0..floor(n/2)} (-4)^k * binomial(-7/2,k) * binomial(k,n-2*k).

A361488 Diagonal of rational function 1/(1 - (x^3 + y^3 + x^4*y)).

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 6, 12, 6, 20, 60, 60, 90, 280, 420, 532, 1330, 2520, 3444, 6804, 14112, 21912, 37884, 77616, 133914, 223080, 432432, 793364, 1341912, 2471040, 4629196, 8076640, 14453010, 26960232, 48308832, 85794852, 157947816, 287413152, 512697900, 933072064

Offset: 0

Seiichi Manyama, Mar 22 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A006139, A115962.

Cf. A360267, A361727.

Table[Sum[Binomial[2*k,k] * Binomial[k,n-3*k], {k,0,n/3}], {n,0,20}] (* Vaclav Kotesovec, Mar 23 2023 *)

a(n) = sum(k=0, n\3, binomial(2*k, k)*binomial(k, n-3*k));

G.f.: 1/sqrt(1 - 4 * x^3 * (1+x)).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(k,n-3*k).
From Vaclav Kotesovec, Mar 23 2023: (Start)
Recurrence: n*a(n) = 2*(2*n-3)*a(n-3) + 4*(n-2)*a(n-4).
a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 1.835086681639635368143322042736678753... is the positive real root of the equation d^4 - 4*d - 4 = 0 and c = 0.2982650309662120181812121016104223... is the largest real root of the equation 1 - 20*c + 132*c^2 - 364*c^3 + 364*c^4 = 0. (End)

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A128063 Hankel transform of A115962.

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A115951 Expansion of 1/sqrt(1-4*x*y-4*x^2*y).

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A377186 Expansion of 1/(1 - 4*x^2 - 4*x^3)^(3/2).

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A361726 Diagonal of rational function 1/(1 - (1 + x*y) * (x^2 + y^2)).

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A377189 Expansion of 1/(1 - 4*x^2 - 4*x^3)^(5/2).

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A377190 Expansion of 1/(1 - 4*x^2 - 4*x^3)^(7/2).

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A361488 Diagonal of rational function 1/(1 - (x^3 + y^3 + x^4*y)).

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Mathematica

PARI

Formula

A115951 Expansion of 1/sqrt(1-4xy-4x^2y).

A377186 Expansion of 1/(1 - 4x^2 - 4x^3)^(3/2).

A361726 Diagonal of rational function 1/(1 - (1 + xy) (x^2 + y^2)).

A377189 Expansion of 1/(1 - 4x^2 - 4x^3)^(5/2).

A377190 Expansion of 1/(1 - 4x^2 - 4x^3)^(7/2).