A002422 -id:A002422 - cp's OEIS frontend

A182411 Triangle T(n,k) = (2k)!(2n)!/(k!n!*(k+n)!) with k=0..n, read by rows.

1, 2, 2, 6, 4, 6, 20, 10, 12, 20, 70, 28, 28, 40, 70, 252, 84, 72, 90, 140, 252, 924, 264, 198, 220, 308, 504, 924, 3432, 858, 572, 572, 728, 1092, 1848, 3432, 12870, 2860, 1716, 1560, 1820, 2520, 3960, 6864, 12870, 48620, 9724, 5304, 4420, 4760, 6120, 8976

Offset: 0

Bruno Berselli, Apr 27 2012

This is a companion to the triangle A068555.
Row sum is 2*A132310(n-1) + A000984(n) for n>0, where A000984(n) = T(n,0) = T(n,n). Also:
T(n,1)  = -A002420(n+1).
T(n,2)  =  A002421(n+2).
T(n,3)  = -A002422(n+3) = 2*A007272(n).
T(n,4)  =  A002423(n+4).
T(n,5)  = -A002424(n+5).
T(n,6)  =  A020923(n+6).
T(n,7)  = -A020925(n+7).
T(n,8)  =  A020927(n+8).
T(n,9)  = -A020929(n+9).
T(n,10) =  A020931(n+10).
T(n,11) = -A020933(n+11).

			Triangle begins:
      1;
      2,    2;
      6,    4,    6;
     20,   10,   12,   20;
     70,   28,   28,   40,   70;
    252,   84,   72,   90,  140,  252;
    924,  264,  198,  220,  308,  504,  924;
   3432,  858,  572,  572,  728, 1092, 1848,  3432;
  12870, 2860, 1716, 1560, 1820, 2520, 3960,  6864, 12870;
  48620, 9724, 5304, 4420, 4760, 6120, 8976, 14586, 25740, 48620;
  ...
Sum_{k=0..8} T(8,k) = 12870 + 2860 + 1716 + 1560 + 1820 + 2520 + 3960 + 6864 + 12870 = 2*A132310(7) + A000984(8) = 2*17085 + 12870 = 47040.

Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third edition), page 11.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 103.

Alexander Borisov, Quotient singularities, integer ratios of factorials and the Riemann Hypothesis, arXiv:math/0505167 [math.NT], 2005; International Mathematics Research Notices, Vol. 2008, Article ID rnn052, page 2 (Theorem 2).
Ira Gessel, Integer quotients of factorials and algebraic multivariable hypergeometric series, MIT Combinatorics Seminar, September 2011 (slides).
Hans-Christian Herbig and Mateus de Jesus Gonçalves, On the numerology of trigonometric polynomials, arXiv:2311.13604 [math.HO], 2023.
Kevin Limanta and Norman Wildberger, Super Catalan Numbers, Chromogeometry, and Fourier Summation over Finite Fields, arXiv:2108.10191 [math.CO], 2021. See Table 1 p. 2 where terms are shown as an array.

Cf. A000984, A002420-A020933, A068555, A132310.

[Factorial(2*k)*Factorial(2*n)/(Factorial(k)*Factorial(n)*Factorial(k+n)): k in [0..n], n in [0..9]];

Flatten[Table[Table[(2 k)! ((2 n)!/(k! n! (k + n)!)), {k, 0, n}], {n, 0, 9}]]

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

Original entry on oeis.org

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]]

cp's OEIS Frontend

A182411 Triangle T(n,k) = (2k)!(2n)!/(k!n!*(k+n)!) with k=0..n, read by rows.

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

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Mathematica

Formula

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

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Mathematica

cp's OEIS Frontend

A182411 Triangle T(n,k) = (2*k)!*(2*n)!/(k!*n!*(k+n)!) with k=0..n, read by rows.

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Magma

Mathematica

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A182411 Triangle T(n,k) = (2k)!(2n)!/(k!n!*(k+n)!) with k=0..n, read by rows.