A020925 -id:A020925 - cp's OEIS frontend

A020927 Expansion of (1-4*x)^(15/2).

1, -30, 390, -2860, 12870, -36036, 60060, -51480, 12870, 2860, 1716, 1560, 1820, 2520, 3960, 6864, 12870, 25740, 54340, 120120, 276276, 657800, 1614600, 4071600, 10518300, 27768312, 74760840, 204900080, 570793080, 1613966640, 4626704368, 13432367520

Offset: 0

N. J. A. Sloane

sign

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

CoefficientList[Series[(1-4x)^(15/2),{x,0,30}],x] (* Harvey P. Dale, Oct 03 2012 *)

D-finite with recurrence: n*a(n) +2*(-2*n+17)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
From Amiram Eldar, Mar 25 2022: (Start)
a(n) = (-4)^n*binomial(15/2, n).
Sum_{n>=0} 1/a(n) = 972/1001 + 34*Pi/(3^10*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 18235778692/17595703125 - 68*log(phi)/(5^9*sqrt(5)), where phi is the golden ratio (A001622). (End)

A020929 Expansion of (1-4*x)^(17/2).

Original entry on oeis.org

1, -34, 510, -4420, 24310, -87516, 204204, -291720, 218790, -48620, -9724, -5304, -4420, -4760, -6120, -8976, -14586, -25740, -48620, -97240, -204204, -447304, -1016600, -2386800, -5768100, -14304888, -36312408, -94143280, -248807240, -669205680, -1829162192

Offset: 0

N. J. A. Sloane

sign

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

CoefficientList[Series[(1 - 4 x)^(17/2), {x, 0, 33}], x] (* Vincenzo Librandi, Jan 18 2020 *)

D-finite with recurrence: n*a(n) +2*(-2*n+19)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
From Amiram Eldar, Mar 25 2022: (Start)
a(n) = (-4)^n*binomial(17/2, n).
Sum_{n>=0} 1/a(n) = 49600/51051 - 38*Pi/(3^11*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 1542987607648/1495634765625 - 76*log(phi)/(5^10*sqrt(5)), where phi is the golden ratio (A001622). (End)

A020931 Expansion of (1-4*x)^(19/2).

Original entry on oeis.org

1, -38, 646, -6460, 41990, -184756, 554268, -1108536, 1385670, -923780, 184756, 33592, 16796, 12920, 12920, 15504, 21318, 32604, 54340, 97240, 184756, 369512, 772616, 1679600, 3779100, 8767512, 20907144, 51106352, 127765880, 326023280, 847660528, 2242198816

Offset: 0

N. J. A. Sloane

sign

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

CoefficientList[Series[(1-4x)^(19/2),{x,0,30}],x] (* Harvey P. Dale, Jul 03 2013 *)

D-finite with recurrence: n*a(n) +2*(-2*n+21)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
From Amiram Eldar, Mar 25 2022: (Start)
a(n) = (-4)^n*binomial(19/2, n).
Sum_{n>=0} 1/a(n) = 45052/46189 + 14*Pi/(3^11*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 6955761045148/6765966796875 - 84*log(phi)/(5^11*sqrt(5)), where phi is the golden ratio (A001622). (End)

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

Original entry on oeis.org

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)

cp's OEIS Frontend

A020927 Expansion of (1-4*x)^(15/2).

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A020929 Expansion of (1-4*x)^(17/2).

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A020931 Expansion of (1-4*x)^(19/2).

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