Detlef Meya - cp's OEIS frontend

A380206 Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. 5 Pi/3} (negated).

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

Offset: 1

Detlef Meya, Jan 16 2025

			-4.841900132896448626653713755364830580644916393751353477278...

Jonathan M. Borwein and Armin Straub, Special Values of Generalized Log-sine Integrals, ISSAC '11: Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, 2011, pp. 43-50.
Armin Straub, A Mathematica package for evaluating log-sine integrals

Cf. A379042, A379273, A380205.

RealDigits[(1/108)*(-11*Pi^3 + 24*Sqrt[3]*Pi^2*Log[3/2] - 180*Pi*Log[3/2]^2 - 36*Sqrt[3]*Log[3/2]*PolyGamma[1, 1/3]), 10, 100] // First

-Integral_{0..5*Pi/3} log(3*sin(x/2))^2 dx = (1/108)*(-11*Pi^3 + 24*Sqrt(3)*Pi^2*Log(3/2) - 180*Pi*Log(3/2)^2 - 36*Sqrt(3)*Log(3/2)*PolyGamma(1, 1/3)).

Equals (-Integral_{0..2 Pi} log(3*sin(x/2))^2 dx) - (-Integral_{0..Pi/3} log(3*sin(x/2))^2 dx). (This formula was suggested by Mathematica.)

A380205 Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. 4 Pi/3} (negated).

Original entry on oeis.org

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

Offset: 1

Detlef Meya, Jan 16 2025

			-4.260288739151063174322652953730500534988887875869780138153916257727...

Jonathan M. Borwein and Armin Straub, Special Values of Generalized Log-sine Integrals, ISSAC '11: Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, 2011, pp. 43-50.
Armin Straub, A Mathematica package for evaluating log-sine integrals

Cf. A379042, A379273, A380206.

Digits:= 100: evalf(Int(log(3*sin(x/2))^2, x = 0..4*Pi/3)); # Peter Luschny, Jan 28 2025

NIntegrate[Log[3*Sin[x/2]]^2, {x, 2*Pi/3, 2*Pi}, WorkingPrecision -> 100]

A379273 Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. 2*Pi/3} (negated).

Original entry on oeis.org

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

Offset: 1

Detlef Meya, Dec 19 2024

			-1.9403919820720596979364925591310637161191841878362545269432607629448...

Jonathan M. Borwein and Armin Straub, Special Values of Generalized Log-sine Integrals, ISSAC '11: Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, 2011, pp. 43-50.
Armin Straub, A Mathematica package for evaluating log-sine integrals

Cf. A379042.

RealDigits[(1/162)*(-4*Pi^3 + 324*Im[PolyLog[3, 1 - (-1)^(2/3)]] -
   108*Pi*Log[3/2]^2 + 27*Pi*Log[3]^2 + 12*Sqrt[3]*Pi^2*Log[27/4] -
   18*Sqrt[3]*Log[27/4]*PolyGamma[1, 2/3])
, 10, 105] // First

-Integral_{0..2*Pi/3} log(3*sin(x/2))^2 dx = (1/162)*(-4*Pi^3 + 324*Im(PolyLog(3, 1 - (-1)^(2/3))) -
   108*Pi*Log(3/2)^2 + 27*Pi*Log(3)^2 + 12*Sqrt(3)*Pi^2*Log(27/4) -
   18*Sqrt(3)*Log(27/4)*PolyGamma(1, 2/3)). (This formula was suggested by Mathematica.)

A379042 Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. Pi/3} (negated).

Original entry on oeis.org

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

Offset: 1

Detlef Meya, Dec 14 2024

			-1.3587805883266742456054317574967336704631556699546811878188991347065...

Jonathan M. Borwein and Armin Straub, Special Values of Generalized Log-sine Integrals, ISSAC '11: Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, 2011, pp. 43-50.
Armin Straub, A Mathematica package for evaluating log-sine integrals

Cf. A258759 (Ls_3(Pi/3)).

Digits:= 106: evalf(Int(log(3*sin(x/2))^2, x = 0..Pi/3)); # Peter Luschny, Dec 16 2024

RealDigits[-((7*Pi^3)/108) - (2*Pi^2*Log[3/2])/(3*Sqrt[3]) - (1/3)*Pi* Log[2]^2 + (2/3)*Pi*Log[2]*Log[3] - (1/3)*Pi*Log[3]^2 + (Log[3/2]*PolyGamma[1, 1/3])/Sqrt[3], 10, 105] // First

-Integral_{0..Pi/3} log(3*sin(x/2))^2 dx = -((7*Pi^3)/108) - (2*Pi^2*Log(3/2))/(3*Sqrt(3)) - (1/3)*Pi*Log(2)^2 + (2/3)*Pi*Log(2)*Log(3) - (1/3)*Pi*Log(3)^2 + (Log(3/2)*PolyGamma(1, 1/3))/Sqrt(3). (This formula was suggested by Mathematica.)

A375855 Triangle read by rows: T(n, k) = 2^k * hypergeom([-n, -k], [], -1/2).

Original entry on oeis.org

1, 1, 1, 1, 0, -2, 1, -1, -2, 2, 1, -2, 0, 8, 8, 1, -3, 4, 8, -24, -88, 1, -4, 10, -4, -56, 32, 592, 1, -5, 18, -34, -40, 312, 400, -3344, 1, -6, 28, -88, 96, 512, -1472, -6144, 14464, 1, -7, 40, -172, 448, 32, -4544, 4160, 63616, -2944

Offset: 0

Detlef Meya, Aug 31 2024

			Triangle starts:
[0] 1;
[1] 1,  1;
[2] 1,  0, -2;
[3] 1, -1, -2,    2;
[4] 1, -2,  0,    8,   8;
[5] 1, -3,  4,    8, -24, -88;
[6] 1, -4, 10,   -4, -56,  32,   592;
[7] 1, -5, 18,  -34, -40, 312,   400, -3344;
[8] 1, -6, 28,  -88,  96, 512, -1472, -6144, 14464;
[9] 1, -7, 40, -172, 448,  32, -4544,  4160, 63616, -2944;
...

John Tyler Rascoe, Rows n = 0..140, flattened

Cf. A375854, A000012, A295382 (main diagonal).

T := (n, k) -> 2^k * hypergeom([-n, -k], [], -1/2);
for n from 0 to 9 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, Sep 02 2024

T[n_, k_] := (-1)^k*Sum[(-2)^(k - j)*Binomial[n, j]*Binomial[k, j]*j!, {j, 0, k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

from math import comb, factorial
def A375855(n,k):
    return (-1)**k*sum((-2)**(k-j)*comb(n, j)*comb(k, j)*factorial(j) for j in range(k+1)) # John Tyler Rascoe, Sep 05 2024

T(n, k) = (-1)^k*Sum_{j=0..k} (-2)^(k - j)*binomial(n, j)*binomial(k, j)*j!.

A375854 Triangle read by rows: T(n, k) = 2^k * hypergeom([-n, -k], [], 1/2).

Original entry on oeis.org

1, 1, 3, 1, 4, 14, 1, 5, 22, 86, 1, 6, 32, 152, 648, 1, 7, 44, 248, 1256, 5752, 1, 8, 58, 380, 2248, 12032, 58576, 1, 9, 74, 554, 3768, 23272, 130768, 671568, 1, 10, 92, 776, 5984, 42112, 270400, 1586944, 8546432, 1, 11, 112, 1052, 9088, 72032, 523072, 3479744, 21241984, 119401856

Offset: 0

Detlef Meya, Aug 31 2024

			Triangle starts:
[0] 1;
[1] 1, 3;
[2] 1, 4, 14;
[3] 1, 5, 22, 86;
[4] 1, 6, 32, 152, 648;
[5] 1, 7, 44, 248, 1256, 5752;
[6] 1, 8, 58, 380, 2248, 12032, 58576;
[7] 1, 9, 74, 554, 3768, 23272, 130768, 671568;
[8] 1, 10, 92, 776, 5984, 42112, 270400, 1586944, 8546432;
[9] 1, 11, 112, 1052, 9088, 72032, 523072, 3479744, 21241984, 119401856;
...

Cf. A375855, A000012, A087912 (main diagonal).

T := (n, k) -> 2^k * hypergeom([-n, -k], [], 1/2):
for n from 0 to 9 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, Sep 02 2024

T[n_, k_] := Sum[2^(k - j)*Binomial[n, j]*Binomial[k, j]*j!, {j, 0, k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

from math import isqrt, comb, factorial
def A375854(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    b = n-comb(a+1,2)
    return sum(comb(a,j)*comb(b,j)*factorial(j)<Chai Wah Wu, Nov 13 2024

T(n, k) = Sum_{j=0..k} 2^(k - j)*binomial(n, j)*binomial(k, j)*j!.

A375613 Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], 1/4).

Original entry on oeis.org

1, 1, 5, 2, 9, 41, 6, 26, 113, 493, 24, 102, 434, 1849, 7889, 120, 504, 2118, 8906, 37473, 157781, 720, 3000, 12504, 52134, 217442, 907241, 3786745, 5040, 20880, 86520, 358584, 1486470, 6163322, 25560529, 106028861, 40320, 166320, 686160, 2831160, 11683224, 48219366, 199040786, 821723673, 3392923553

Offset: 0

Detlef Meya, Aug 21 2024

			Triangle starts:
[0] 1;
[1] 1, 5;
[2] 2, 9, 41;
[3] 6, 26, 113, 493;
[4] 24, 102, 434, 1849, 7889;
[5] 120, 504, 2118, 8906, 37473, 157781;
[6] 720, 3000, 12504, 52134, 217442, 907241, 3786745;
[7] 5040, 20880, 86520, 358584, 1486470, 6163322, 25560529, 106028861;
...

Cf. A375612, A000142, A056545 (main diagonal).

Cf. A374427, A374428, A375446, A375447, A375597, A375600.

T[n_, k_] := Sum[4^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

T(n, k) = Sum_{j=0..k} 4^(k - j)*binomial(k, k - j)*(n - j)!.

A375612 Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], -1/4).

Original entry on oeis.org

1, 1, 3, 2, 7, 25, 6, 22, 81, 299, 24, 90, 338, 1271, 4785, 120, 456, 1734, 6598, 25121, 95699, 720, 2760, 10584, 40602, 155810, 598119, 2296777, 5040, 19440, 75000, 289416, 1117062, 4312438, 16651633, 64309755, 40320, 156240, 605520, 2347080, 9098904, 35278554, 136801778, 530555479, 2057912161

Offset: 0

Detlef Meya, Aug 21 2024

			Triangle starts:
[0] 1;
[1] 1, 3;
[2] 2, 7, 25;
[3] 6, 22, 81, 299;
[4] 24, 90, 338, 1271, 4785;
[5] 120, 456, 1734, 6598, 25121, 95699;
[6] 720, 2760, 10584, 40602, 155810, 598119, 2296777;
[7] 5040, 19440, 75000, 289416, 1117062, 4312438, 16651633, 64309755;
...

Cf. A375613, A000142, A001907 (main diagonal).

Cf. A374427, A374428, A375446, A375447, A375597, A375600.

T[n_, k_] := (-1)^k*Sum[(-4)^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

T(n, k) = (-1)^k*Sum_{j=0..k} (-4)^(k - j)*binomial(k, k - j)*(n - j)!.

A375600 Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], 2/3).

Original entry on oeis.org

1, 1, 5, 2, 8, 34, 6, 22, 82, 314, 24, 84, 296, 1052, 3784, 120, 408, 1392, 4768, 16408, 56792, 720, 2400, 8016, 26832, 90032, 302912, 1022320, 5040, 16560, 54480, 179472, 592080, 1956304, 6474736, 21468848, 40320, 131040, 426240, 1387680, 4521984, 14750112, 48162944, 157438304, 515252608

Offset: 0

Detlef Meya, Aug 20 2024

			Triangle starts:
[0] 1;
[1] 1, 5;
[2] 2, 8, 34;
[3] 6, 22, 82, 314;
[4] 24, 84, 296, 1052, 3784;
[5] 120, 408, 1392, 4768, 16408, 56792;
[6] 720, 2400, 8016, 26832, 90032, 302912, 1022320;
[7] 5040, 16560, 54480, 179472, 592080, 1956304, 6474736, 21468848;
...

Cf. A375597, A000142, A097817 (main diagonal).

Cf. A374427, A374428, A375446, A375447.

T[n_, k_] := 2^k*Sum[(3/2)^(k - j)*Binomial[k, k - j]*((n - j)!), {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

T(n, k) = 2^k*Sum_{j=0..k} (3/2)^(k - j)*binomial(k, k - j)*(n - j)!.

A375597 Triangle read by rows: T(n, k) = n! * 3^k * hypergeom([-k], [-n], -2/3).

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 6, 14, 34, 82, 24, 60, 152, 388, 1000, 120, 312, 816, 2144, 5656, 14968, 720, 1920, 5136, 13776, 37040, 99808, 269488, 5040, 13680, 37200, 101328, 276432, 755216, 2066032, 5659120, 40320, 110880, 305280, 841440, 2321664, 6412128, 17725952, 49045792, 135819136

Offset: 0

Detlef Meya, Aug 20 2024

			Triangle starts:
[0] 1;
[1] 1, 1;
[2] 2, 4, 10;
[3] 6, 14, 34, 82;
[4] 24, 60, 152, 388, 1000;
[5] 120, 312, 816, 2144, 5656, 14968;
[6] 720, 1920, 5136, 13776, 37040, 99808, 269488;
[7] 5040, 13680, 37200, 101328, 276432, 755216, 2066032, 5659120;
...

Cf. A375600, A000142.

Cf. A374427, A374428, A375446, A375447.

T[n_, k_] := (-2)^k*Sum[(-3/2)^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

T(n, k) = (-2)^k*Sum_{j=0..k} (-3/2)^(k - j)*binomial(k, k - j)*(n - j)!.

cp's OEIS Frontend

User: Detlef Meya

A380206 Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. 5 Pi/3} (negated).

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A380205 Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. 4 Pi/3} (negated).

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A379273 Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. 2*Pi/3} (negated).

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A379042 Decimal expansion of the generalized log-sine integral with k = 0, n = 3, m = 3, from {0 .. Pi/3} (negated).

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Maple

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A375855 Triangle read by rows: T(n, k) = 2^k * hypergeom([-n, -k], [], -1/2).

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A375854 Triangle read by rows: T(n, k) = 2^k * hypergeom([-n, -k], [], 1/2).

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Maple

Mathematica

Python

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A375613 Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], 1/4).

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Mathematica

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A375612 Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], -1/4).

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