A002423 -id:A002423 - cp's OEIS frontend

A020933 Expansion of (1-4*x)^(21/2).

1, -42, 798, -9044, 67830, -352716, 1293292, -3325608, 5819814, -6466460, 3879876, -705432, -117572, -54264, -38760, -36176, -40698, -52668, -76076, -120120, -204204, -369512, -705432, -1410864, -2939300, -6348888, -14162904, -32522224, -76659528, -185040240

Offset: 0

N. J. A. Sloane

sign

Cf. A001622, A002420, A002421, A002422, A002423, A002424, A020923, A020925, A020927, A020929, A020931.

CoefficientList[Series[Surd[(1-4x)^21,2],{x,0,30}],x] (* Harvey P. Dale, Feb 25 2020 *)

D-finite with recurrence: n*a(n) +2*(-2*n+23)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
From Amiram Eldar, Mar 25 2022: (Start)
a(n) = (-4)^n*binomial(21/2, n).
Sum_{n>=0} 1/a(n) = 406240/415701 - 46*Pi/(3^13*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 728323714975904/710426513671875 - 92*log(phi)/(5^12*sqrt(5)), where phi is the golden ratio (A001622). (End)

A182534 Array read by antidiagonals: coefficient of the Euler-Mascheroni constant in below expression.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 5, 4, 2, 6, 14, 10, 3, 4, 10, 42, 28, 6, 6, 5, 20, 132, 84, 14, 12, 6, 10, 35, 429, 264, 36, 28, 10, 12, 14, 70, 1430, 858, 99, 72, 20, 20, 14, 28, 126, 4862, 2860, 286, 198, 45, 40, 20, 28, 42, 252

Offset: 1

John M. Campbell, May 05 2012

The (i,j)-entry of the array is the coefficient of the Euler-Mascheroni constant in: -2^(i+2j-1)/Pi*int(log(x)*cos(x)^i*sin(x)^(2j-1)/x, x=0..infinity); see Mathematica code below.
First row:     A000108.
Second row:   -A002420.
Third row:     A007054.
Fourth row:    A002421.
Fifth row:     A007272.
Sixth row:    -A002422.
Eighth row:    A002423.
First column:  A001405.
Second column: A089408.
Odd entries on main diagonal: A126596.

			Evaluate: -256/Pi*int(cos(x)^3*log(x)*sin(x)^5/x, x=0..infinity) = 3*eulergamma-log(9/8). Thus the (3,3) entry of the array is 3, the coefficient of the Euler-Mascheroni constant in this expression.
The array begins as:
| 1   1   2   5   14  42  132 429  ... |
| 2   2   4   10  28  84  264 858  ... |
| 3   2   3   6   14  36  99  286  ... |
| 6   4   6   12  28  72  198 572  ... |
| 10  5   6   10  20  45  110 286  ... |
| 20  10  12  20  40  90  220 572  ... |
| 35  14  14  20  35  70  154 364  ... |
| 70  28  28  40  70  140 308 728  ... |
| ... ... ... ... ... ... ... ...  ... |

Cf. A000108, A002420, A007054, A002421, A007272, A002422, A002423, A001405, A089408, A126596.

A[a_, b_] :=
  A[a, b] =
   Array[Coefficient[
      Integrate[
        Log[x]*Cos[x]^#1*Sin[x]^(2 #2 - 1)/x, {x, 0,
         Infinity}] (2^(#1 + 2 #2 - 1))/(-\[Pi]), EulerGamma] &, {a, b}];
A[11, 11];
Print[A[11, 11] // MatrixForm];
Table2 = {};
k = 1;
While[k < 11, Table1 = {};
  i = 1;
  j = k;
  While[0 < j,
    AppendTo[Table1,
    First[Take[First[Take[A[11, 11], {i, i}]], {j, j}]]];
    j = j - 1;
    i = i + 1];
    AppendTo[Table2, Table1];
    k++];
Print[Flatten[Table2]]

A382874 Expansion of g.f. 2-hypergeom([3/2,7/2],[-1/2],4*x).

Original entry on oeis.org

1, 42, 1890, 32340, 378378, 3567564, 29201172, 216164520, 1484052570, 9607866268, 59342703420, 352648983960, 2029131058500, 11360419371000, 62125264788840, 332868702695760, 1751865025825530, 9075126224864700, 46353422502086700, 233788539957892920

Offset: 0

Karol A. Penson, Apr 07 2025

nonn

Cf. A002423, A001700, A002457, A002421.

seq(coeff(series(2-hypergeom([3/2, 7/2], [-1/2], 4*x), x, k+1), x, k), k=0..19);

my(x='x+O('x^30)); Vec(2 - hypergeom([3/2,7/2],[-1/2],4*x)) \\ Michel Marcus, Apr 07 2025

a(0) = 1, a(n) = 8*4^n*(4*n^2 - 1)*Gamma(7/2 + n)/(15*sqrt(Pi)*n!), n>=1.
G.f.: 2 + (768*x^2 + 64*x - 1)/(1 - 4*x)^(11/2).
For n>=1, a(n) = (2*n-1) * (2*n+1)^2 * (2*n+3) * (2*n+5) * binomial(2*n,n)/15. - Vaclav Kotesovec, Apr 07 2025

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A020933 Expansion of (1-4*x)^(21/2).

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A182534 Array read by antidiagonals: coefficient of the Euler-Mascheroni constant in below expression.

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A382874 Expansion of g.f. 2-hypergeom([3/2,7/2],[-1/2],4*x).

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