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A387237 Expansion of 1/((1-x) * (1-5*x))^(5/2).

1, 15, 145, 1155, 8260, 55188, 351960, 2170080, 13042095, 76827465, 445335891, 2547479025, 14412134100, 80773641900, 449065521300, 2479190589180, 13603361708775, 74238475926825, 403197150223175, 2180369322394725, 11744998515662720, 63044308615576200, 337323759106291100

Offset: 0

Seiichi Manyama, Aug 23 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A026375, A385563, A387238.

Cf. A026377, A374508.

R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-5*x))^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025

CoefficientList[Series[1/((1-x)*(1-5*x))^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1-5*x))^(5/2))

n*a(n) = (6*n+9)*a(n-1) - 5*(n+3)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 5^k * binomial(-5/2,k) * binomial(-5/2,n-k).
a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = Sum_{k=0..n} 4^k * 5^(n-k) * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k) = (binomial(n+4,2)/6) * A026377(n+2).
a(n) = (-1)^n * Sum_{k=0..n} 6^k * (5/6)^(n-k) * binomial(-5/2,k) * binomial(k,n-k).

A387239 a(n) = Sum_{k=0..n} binomial(n+3,k+3) * binomial(2*k+6,k+6).

Original entry on oeis.org

1, 12, 95, 630, 3801, 21672, 119154, 639180, 3369795, 17543196, 90476100, 463291920, 2359240975, 11961944400, 60440659640, 304543085040, 1531044995355, 7682898791700, 38494752520175, 192632866196694, 962948703201331, 4809438625979592, 24002988378037350, 119719958370912900

Offset: 0

Seiichi Manyama, Aug 23 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A026376, A026377.

Cf. A126190.

[&+[Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 24 2025

Table[Sum[Binomial[n+3,k+3]* Binomial[2*k+6, k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 24 2025 *)

a(n) = sum(k=0, n, binomial(n+3, k+3)*binomial(2*k+6, k+6));

n*(n+6)*a(n) = (n+3) * (3*(2*n+5)*a(n-1) - 5*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+3*x+x^2)^(n+3).
E.g.f.: exp(3*x) * BesselI(3, 2*x), with offset 3.

A387272 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+2,k+2) * binomial(2*k+4,k+4).

Original entry on oeis.org

1, 12, 100, 720, 4815, 30884, 193144, 1188576, 7236690, 43741720, 263056728, 1576298464, 9421080123, 56200937940, 334801389360, 1992471776448, 11848869296622, 70425535830696, 418426332826200, 2485390365370080, 14760336569524854, 87650482093915752

Offset: 0

Seiichi Manyama, Aug 24 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A344055, A387273, A387274.

Cf. A387280.

[&+[2^(n-k) * Binomial(n+2,k+2) * Binomial(2*k+4,k+4): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025

Table[Sum[2^(n-k)*Binomial[n+2,k+2]*Binomial[2*k+4,k+4],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)

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

n*(n+4)*a(n) = (n+2) * (4*(2*n+3)*a(n-1) - 12*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+4*x+x^2)^(n+2).
E.g.f.: exp(4*x) * BesselI(2, 2*x), with offset 2.

A387278 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+1,k+1) * binomial(2*k+2,k+2).

Original entry on oeis.org

1, 10, 78, 560, 3885, 26550, 180285, 1221400, 8272251, 56062550, 380361212, 2583867720, 17575724491, 119705522370, 816297170310, 5572945684800, 38088275031435, 260576833989150, 1784382167211378, 12229792774162800, 83888652677196591, 575858959975595010

Offset: 0

Seiichi Manyama, Aug 24 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A098409, A387275, A387276, A387277.

Cf. A026376, A344055.

[&+[3^(n-k) * Binomial(n+1,k+1) * Binomial(2*k+2,k+2): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025

Table[Sum[3^(n-k)*Binomial[n+1,k+1]*Binomial[2*k+2,k+2],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)

a(n) = sum(k=0, n, 3^(n-k)*binomial(n+1, k+1)*binomial(2*k+2, k+2));

n*(n+2)*a(n) = (n+1) * (5*(2*n+1)*a(n-1) - 21*n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+5*x+x^2)^(n+1).
E.g.f.: exp(5*x) * BesselI(1, 2*x), with offset 1.

A387294 Decimal expansion of the largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

Original entry on oeis.org

2, 9, 5, 7, 0, 8, 0, 0, 7, 9, 6, 3, 5, 4, 4, 8, 1, 5, 1, 5, 6, 1, 8, 7, 2, 5, 8, 1, 3, 4, 5, 0, 3, 7, 6, 5, 3, 0, 5, 1, 8, 0, 8, 7, 0, 0, 4, 0, 8, 9, 9, 7, 9, 2, 3, 0, 0, 0, 5, 1, 8, 7, 0, 3, 7, 2, 7, 8, 5, 7, 5, 7, 7, 5, 3, 2, 0, 1, 3, 8, 4, 9, 7, 2, 2, 0, 0, 6, 3, 9

Offset: 1

Paolo Xausa, Aug 25 2025

This is the dihedral angle between a triangular face in the antiprism part of the solid and a triangular face in the cupola part of the solid.

Also the analogous dihedral angle in a gyroelongated triangular bicupola (Johnson solid J_44).

			2.9570800796354481515618725813450376530518087004...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated triangular bicupola.
Wikipedia, Gyroelongated triangular cupola.

Cf. other J_22 dihedral angles: A195698, A387295, A387296, A387297.
Cf. A344076 (J_22 volume), A344077 (J_22 surface area).
Cf. A385256 (J_44 volume), A385257 (J_44 surface area).
Cf. A137914, A246724.

First[RealDigits[ArcSec[3] + ArcCos[1 - Sqrt[12]/3], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["J22", "DihedralAngles"]], 10, 100]]

Equals arccos(1/3) + arccos(1 - 2*sqrt(3)/3) = A137914 + arccos(-A246724).

A387295 Decimal expansion of the second largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

Original entry on oeis.org

2, 6, 8, 1, 4, 3, 7, 2, 8, 0, 4, 1, 9, 1, 8, 2, 7, 4, 7, 5, 9, 0, 8, 0, 0, 5, 0, 5, 6, 1, 2, 8, 0, 8, 0, 3, 1, 5, 8, 4, 8, 8, 3, 3, 8, 6, 0, 6, 3, 9, 0, 8, 5, 7, 4, 9, 0, 4, 6, 6, 8, 4, 9, 9, 3, 8, 5, 7, 7, 7, 3, 0, 8, 9, 5, 7, 7, 3, 4, 2, 1, 7, 2, 5, 6, 1, 4, 6, 3, 8

Offset: 1

Paolo Xausa, Aug 25 2025

This is the dihedral angle between a triangular face in the antiprism part of the solid and a square face in the cupola part of the solid.

Also the analogous dihedral angle in a gyroelongated triangular bicupola (Johnson solid J_44).

			2.6814372804191827475908005056128080315848833860639...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated triangular bicupola.
Wikipedia, Gyroelongated triangular cupola.

Cf. other J_22 dihedral angles: A195698, A387294, A387296, A387297.
Cf. A344076 (J_22 volume), A344077 (J_22 surface area).
Cf. A385256 (J_44 volume), A385257 (J_44 surface area).
Cf. A195696, A246724.

First[RealDigits[ArcTan[Sqrt[2]] + ArcCos[1 - Sqrt[12]/3], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J22", "DihedralAngles"]], 2], 10, 100]]

Equals arccos(sqrt(3)/3) + arccos(1 - 2*sqrt(3)/3) = A195696 + arccos(-A246724).

A387296 Decimal expansion of the third largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

Original entry on oeis.org

2, 5, 3, 4, 6, 0, 0, 1, 4, 9, 7, 1, 5, 1, 2, 6, 1, 9, 3, 0, 9, 1, 5, 0, 2, 8, 1, 0, 2, 1, 0, 2, 1, 0, 7, 0, 2, 1, 4, 9, 8, 3, 0, 3, 2, 9, 1, 9, 3, 5, 1, 5, 3, 6, 3, 6, 8, 8, 4, 3, 4, 6, 4, 6, 4, 1, 3, 6, 2, 5, 9, 5, 0, 3, 8, 5, 3, 4, 7, 9, 8, 9, 3, 8, 8, 4, 6, 2, 6, 1

Offset: 1

Paolo Xausa, Aug 26 2025

This is the dihedral angle between triangular faces in the antiprism part of the solid.

Also the analogous dihedral angle in a gyroelongated triangular bicupola (Johnson solid J_44).

			2.5346001497151261930915028102102107021498303291935...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated triangular bicupola.
Wikipedia, Gyroelongated triangular cupola.

Cf. other J_22 dihedral angles: A195698, A387294, A387295, A387297.
Cf. A344076 (J_22 volume), A344077 (J_22 surface area).
Cf. A385256 (J_44 volume), A385257 (J_44 surface area).
Cf. A010469.

First[RealDigits[ArcCos[(1 - Sqrt[12])/3], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J22", "DihedralAngles"]], 3], 10, 100]]

Equals arccos((1 - 2*sqrt(3))/3) = arccos((1 - A010469)/3).

A387297 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

Original entry on oeis.org

1, 7, 2, 6, 1, 2, 0, 6, 6, 2, 2, 9, 4, 6, 7, 3, 4, 6, 9, 4, 2, 6, 9, 4, 3, 4, 0, 3, 0, 9, 7, 0, 5, 0, 2, 7, 7, 3, 4, 1, 4, 6, 8, 6, 9, 1, 0, 5, 3, 9, 0, 3, 0, 8, 3, 9, 4, 4, 9, 7, 0, 3, 7, 0, 0, 6, 3, 8, 6, 5, 2, 6, 3, 0, 5, 3, 7, 5, 7, 7, 6, 1, 8, 6, 8, 7, 5, 4, 7, 7

Offset: 1

Paolo Xausa, Aug 26 2025

This is the dihedral angle between a triangular face and the hexagonal face.

			1.7261206622946734694269434030970502773414686910539...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated triangular cupola.

Cf. other J_22 dihedral angles: A195698, A387294, A387295, A387296.
Cf. A344076 (J_22 volume), A344077 (J_22 surface area).
Cf. A010469.

First[RealDigits[ArcCos[1 - 2/Sqrt[3]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J22", "DihedralAngles"]], 10, 100]]

Equals arccos(1 - 2*sqrt(3)/3) = arccos(1 - A010469/3).

A387311 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k+3) * binomial(2*k+6,k+6).

Original entry on oeis.org

1, 28, 535, 8750, 132041, 1900808, 26557986, 363716220, 4912064355, 65673861484, 871539802276, 11501122783696, 151118588963615, 1978948331160080, 25846338449608184, 336857447941007280, 4382848524348689883, 56947000383926523780, 739095412895790074215, 9583718189242229830798

Offset: 0

Seiichi Manyama, Aug 25 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A387309, A387310.

[&+[3^k * Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025

Table[Sum[3^k * Binomial[n+3,k+3]*Binomial[2*k+6, k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)

a(n) = sum(k=0, n, 3^k*binomial(n+3, k+3)*binomial(2*k+6, k+6));

n*(n+6)*a(n) = (n+3) * (7*(2*n+5)*a(n-1) - 13*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+7*x+9*x^2)^(n+3).
E.g.f.: exp(7*x) * BesselI(3, 6*x) / 27, with offset 3.

A387320 Decimal expansion of the largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

2, 6, 8, 7, 1, 5, 0, 5, 0, 5, 6, 3, 7, 0, 7, 0, 6, 2, 2, 0, 5, 8, 2, 3, 7, 6, 7, 1, 0, 3, 4, 2, 1, 7, 8, 7, 2, 4, 0, 8, 0, 9, 4, 2, 4, 3, 7, 8, 8, 1, 6, 0, 5, 3, 3, 1, 8, 5, 9, 1, 6, 8, 3, 2, 2, 7, 7, 2, 3, 2, 9, 7, 1, 2, 7, 7, 5, 0, 1, 0, 3, 2, 5, 2, 6, 9, 7, 3, 5, 8

Offset: 1

Paolo Xausa, Aug 27 2025

This is the dihedral angle between triangular faces in the antiprism part of the solid.

Also the analogous dihedral angle in a gyroelongated square bicupola (Johnson solid J_45).

			2.687150505637070622058237671034217872408094243788...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Gyroelongated square bicupola.
Polytope Wiki, Gyroelongated square cupola.
Wikipedia, Gyroelongated square bicupola.
Wikipedia, Gyroelongated square cupola.

Cf. other J_23 dihedral angles: A177870, A195702, A387321, A387322, A387323.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A385258 (J_45 volume), A385259 (J_45 surface area).
Cf. A002193.

First[RealDigits[ArcCos[(1 - Sqrt[8 + Sqrt[32]])/3], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["J23", "DihedralAngles"]], 10, 100]]

Equals arccos((1 - 2*sqrt(2 + sqrt(2)))/3) = arccos((1 - 2*sqrt(2 + A002193))/3).

cp's OEIS Frontend

A387237 Expansion of 1/((1-x) * (1-5*x))^(5/2).

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A387239 a(n) = Sum_{k=0..n} binomial(n+3,k+3) * binomial(2*k+6,k+6).

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A387272 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+2,k+2) * binomial(2*k+4,k+4).

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A387278 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+1,k+1) * binomial(2*k+2,k+2).

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A387294 Decimal expansion of the largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

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A387295 Decimal expansion of the second largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

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A387296 Decimal expansion of the third largest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

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A387297 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated triangular cupola (Johnson solid J_22).

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