A001405 -id:A001405 - cp's OEIS frontend

A103769 Trinomial transform of central binomial coefficients A001405.

1, 4, 21, 123, 757, 4788, 30817, 200784, 1320093, 8740284, 58193673, 389233287, 2613338091, 17602627006, 118892784555, 804951501469, 5461228061541, 37120212399708, 252720891884473, 1723088114793535, 11763751150648785

Offset: 0

Paul Barry, Feb 15 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A082760.

CoefficientList[Series[((3*x+1-(21*x^2-10*x+1)^(1/2))/(2*x*(3*x-4)*(7*x-1)))^(1/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

a(n) = sum_{k=0..2n} T(n,k)*C(k,floor(k/2)), where T(n,k) is given by A027907.
a(n) = sum_{k=0..n} sum_{j=0..n} C(n,j)*C(j,k)*C(j+k,floor((j+k)/2)).
G.f.: ((3*x+1-(21*x^2-10*x+1)^(1/2))/(2*x*(3*x-4)*(7*x-1)))^(1/2). - Mark van Hoeij, Nov 16 2011
Conjecture: n*(2n+1)*a(n) +2(-61n^2+57n-20)*a(n-1) +3*(205n^2-523*n+346) * a(n-2) -72*(n-2)*(16n-33)*a(n-3) +567*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Dec 14 2011
a(n) ~ 7^(n+1/2)/sqrt(5*Pi*n). - Vaclav Kotesovec, Oct 24 2012

A134184 A transform of the central binomial coefficients A001405.

Original entry on oeis.org

1, 1, 3, 7, 17, 43, 109, 279, 721, 1871, 4881, 12783, 33585, 88495, 233745, 618719, 1640833, 4358719, 11595841, 30890751, 82391297, 219995007, 588004737, 1573072383, 4211960065, 11286490879, 30265474305

Offset: 0

Paul Barry, Oct 11 2007

Hankel transform is A134185.

G.f.: g(x(1+x)) where g(x) is the g.f. of A001405; a(n)=sum{k=0..n, binomial(k,n-k)*binomial(k,floor(k/2))}; a(n)=sum{k=0..floor(n/2), binomial(n-k,k)*binomial(n-k,floor((n-k)/2))};

A134185 Hankel transform of a transform of the central binomial coefficients A001405.

Original entry on oeis.org

1, 2, 0, -8, -32, -256, 0, 16384, 262144, 8388608, 0, -8589934592, -549755813888, -70368744177664, 0, 1152921504606846976, 295147905179352825856, 151115727451828646838272, 0, -39614081257132168796771975168

Offset: 0

Paul Barry, Oct 11 2007

Hankel transform of A134184.

Cf. A001405, A134184.

a(n) = ((1/2 - 1/sqrt(2))*cos(3*Pi*n/4) + (1/2 + 1/sqrt(2))*cos(Pi*n/4))*2^ceiling(binomial(n+1,2)/2).

A161240 Number of partitions of n into central binomial coefficients A001405.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 12, 15, 19, 22, 29, 33, 40, 47, 56, 63, 76, 85, 100, 113, 131, 146, 169, 187, 214, 237, 268, 295, 334, 365, 410, 449, 499, 545, 606, 657, 727, 789, 868, 940, 1033, 1114, 1219, 1315, 1433, 1542, 1678, 1800, 1954, 2095, 2266, 2426, 2619, 2798

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

			a(6)=8 because we have 6, 33, 321, 3111, 222, 2211, 21111, and 111111. - _Emeric Deutsch_, Jun 21 2009

R. H. Hardin, Table of n, a(n) for n = 1..1000

g := 1/(product(1-x^binomial(j, floor((1/2)*j)), j = 1 .. 15)): gser := series(g, x = 0, 63): seq(coeff(gser, x, n), n = 1 .. 55); # Emeric Deutsch, Jun 21 2009

G.f.: 1/Product_{j>=1} (1 - x*binomial(j, floor(j/2))). - Emeric Deutsch, Jun 21 2009

A161241 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 2 times.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 2, 5, 5, 7, 6, 12, 9, 14, 15, 19, 18, 28, 24, 34, 35, 43, 42, 59, 53, 70, 72, 86, 86, 112, 105, 132, 136, 158, 160, 198, 191, 231, 238, 273, 277, 332, 326, 384, 394, 447, 455, 532, 529, 610, 626, 703, 717, 821, 824, 934, 959, 1066, 1089, 1230, 1240, 1388

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A001405.

A161242 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 3 times.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 2, 3, 3, 3, 6, 5, 6, 9, 8, 9, 14, 12, 14, 19, 19, 21, 28, 26, 30, 37, 38, 41, 53, 50, 56, 68, 69, 74, 91, 88, 98, 115, 118, 125, 150, 149, 163, 188, 192, 204, 239, 239, 260, 293, 303, 322, 367, 372, 400, 444, 462, 487, 548, 557, 597, 657, 682, 718, 795, 812, 868

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A001405.

A161243 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 4 times.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 4, 2, 4, 4, 6, 5, 8, 7, 10, 9, 12, 11, 16, 14, 18, 18, 23, 23, 29, 28, 34, 34, 41, 41, 51, 49, 58, 59, 70, 69, 83, 81, 95, 96, 110, 110, 130, 127, 147, 147, 169, 168, 196, 195, 221, 224, 253, 254, 291, 289, 326, 330, 370, 374, 420, 423, 471, 480, 531, 537

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A001405.

A161244 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 5 times.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 3, 3, 5, 4, 6, 7, 7, 8, 10, 10, 11, 13, 13, 14, 18, 17, 19, 22, 23, 26, 31, 31, 34, 38, 41, 44, 51, 51, 56, 62, 66, 70, 80, 80, 89, 96, 101, 107, 120, 122, 132, 143, 149, 157, 176, 176, 190, 204, 213, 226, 248, 252, 271, 290, 305, 319, 348, 355

Offset: 1

R. H. Hardin Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

A161245 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 6 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 2, 4, 4, 5, 4, 8, 6, 8, 9, 10, 9, 14, 11, 14, 15, 16, 15, 22, 18, 22, 24, 26, 26, 35, 32, 37, 40, 43, 44, 55, 52, 59, 62, 67, 68, 83, 79, 89, 93, 100, 101, 122, 116, 129, 135, 144, 145, 172, 165, 182, 190, 202, 202, 236, 226, 249, 259, 275, 276

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A001405.

A161246 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 7 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 3, 3, 3, 5, 4, 5, 6, 7, 7, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 22, 24, 26, 28, 30, 34, 36, 39, 42, 45, 48, 53, 56, 61, 64, 69, 72, 79, 82, 89, 94, 100, 104, 114, 117, 126, 133, 142, 146, 159, 163, 174, 183, 193, 201, 216, 222

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A001405.

cp's OEIS Frontend

A103769 Trinomial transform of central binomial coefficients A001405.

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A134184 A transform of the central binomial coefficients A001405.

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A134185 Hankel transform of a transform of the central binomial coefficients A001405.

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A161240 Number of partitions of n into central binomial coefficients A001405.

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A161241 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 2 times.

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A161242 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 3 times.

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A161243 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 4 times.

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A161244 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 5 times.

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A161245 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 6 times.

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A161246 Number of partitions of n into central binomial coefficients A001405 where every part appears at least 7 times.

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