A002596 -id:A002596 - cp's OEIS frontend

A001795 Coefficients of Legendre polynomials.

1, 1, 7, 33, 715, 4199, 52003, 334305, 17678835, 119409675, 1641030105, 11435320455, 322476036831, 2295919134019, 32968493968795, 238436656380769, 27767032438524099, 203236010537432691, 2989949596465113373

Offset: 0

N. J. A. Sloane

nonn

Numerators in expansion of sqrt(c(x)), c(x) the g.f. of A000108. - Paul Barry, Jul 12 2005

Coefficient of Legendre_0(x) when x^n is written in term of Legendre polynomials. - Michel Marcus, May 28 2013

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.

Divisor of A048990 and A065097.
Apparently a bisection of A002596.
Bisection of A099024.
Cf. A000108, A001791.

A001795:= func< n | Numerator(Catalan(2*n)/4^n) >;
[A001795(n): n in [0..25]]; // G. C. Greubel, Apr 22 2025

Table[Numerator[CatalanNumber[2*n]/4^n], {n,0,30}] (* G. C. Greubel, Apr 22 2025 *)

my(x='x+O('x^30)); apply(numerator, Vec(((1-sqrt(1-4*x))/(2*x))^(1/2))) \\ Michel Marcus, Feb 04 2022

a(n)=numerator(binomial(2*n-1/2, n)/(2*n+1)) \\ Tani Akinari, Oct 22 2024

def A001795(n): return numerator(catalan_number(2*n)/4^n)
print([A001795(n) for n in range(31)]) # G. C. Greubel, Apr 22 2025

1/(sqrt(1-x) + sqrt(1+x)) = Sum_{n>=0} (a(n)/b(n))*x^(2*n) where b(n) is a power of 2. - Benoit Cloitre, Mar 12 2002
For n >= 1, 2^(n+1)*a(2^(n-1)) = A001791(2^n). - Vladimir Shevelev, Sep 05 2010
a(n) = numerator(binomial(2*n-1/2, n)/(2*n+1)). - Tani Akinari, Oct 22 2024
a(n) = numerator( A000108(2*n)/4^n ). - G. C. Greubel, Apr 22 2025

More terms from Benoit Cloitre, Mar 12 2002

A224270 Absolute values of the numerators of the third column of ( 0 followed by (interleave 0 , A001803(n))/A060818(n) ) and its successive differences.

Original entry on oeis.org

1, 1, 5, 11, 95, 203, 861, 1815, 30459, 63635, 264979, 550069, 4555915, 9412543, 38816525, 79898895, 2627302995, 5392044675, 22104436695, 45256266825, 370241638305, 756514878405, 3088866211275, 6300861570705, 102746354288175, 209286947903319

Offset: 0

Paul Curtz, Apr 02 2013

The array is
  0,          0,      1,     0,   3/2,     0,  15/8, 0,...
  0,          1,     -1,   3/2,  -3/2,  15/8, -15/8,...
  1,         -2,    5/2,    -3,  27/8, -15/4,...
  -3,       9/2,  -11/2,  51/8, -57/8,...
  15/2,     -10,   95/8, -27/2,...
  -35/2,  175/8, -203/8,...
  315/8, -189/4,...
  -693/8,...
Note A001803 in the first column and a variant of A206771(n) in the second column.
Now consider a(n)/A046161(n) and its differences:
  1,            1/2,     5/8,  11/16,  95/128, 203/256, 861/1024,...
  -1/2,         1/8,    1/16,  7/128,  13/256, 49/1024,...   =b(n)/A046161(n)
  5/8,        -1/16,  -1/128, -1/256, -3/1024,...
  -11/16,     7/128,   1/256, 1/1024,...
  95/128,   -13/256, -3/1024,...
  -203/256, 49/1024,...
  861/1024,...
This an autosequence of second kind. The first column is the signed sequence.
(Its companion, the corresponding autosequence of first kind, is 0, 1, 1, 9/8, 5/4,... in A206771).
Main diagonal: 1, 1/8, -1/128,... = A002596(n)/A061549(n) ?
b(n) = a(n+1) - A171977*a(n). Also for two successive rows (with shifted A171977).

			a(n)=numerators of 0+1=1, 0+1/2=1/2, 1/4+3/8=5/8, 3/8+5/16=11/16, 15/32+35/128=95/128,... .

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001790, A001803, A002596, A046161, A061549, A098597, A171977, A206771.

nmax = 25; t1 = Table[ Numerator[ (2*n+1)*(Binomial[2*n, n]/4^n)] / Denominator[ Binomial[2*n, n]/4^n], {n, 0, Ceiling[nmax/2]}]; t2 = Join[{0}, Table[ If[ OddQ[n], 0, t1[[n/2]] ], {n, 1, nmax+2}] ]; t3 = Table[ Differences[t2, n], {n, 0, nmax}]; t3[[All, 3]] // Numerator // Abs (* Jean-François Alcover, Apr 02 2013 *)

Numerators of (0, 0 followed by A001803(n)/(4*A046161(n))) + A001790(n)/A046161(n).

More terms from Jean-François Alcover, Apr 02 2013

A273193 Numerators of the nonzero coefficients in the expansion of 1/hypergeom([Seq_{k=1..m-1} k/m], [], (x/m)^m) for m = 3.

Original entry on oeis.org

1, 2, -16, 904, -25792, 971936, -135875584, 7531512832, -483853915136, 318210896625152, -26070230641872896, 2367374418301892608, -708155254090757373952, 76928188353501090512896, -9044296958948501037252608, 3432739126498593173574909952, -465041552940366454298747600896

Offset: 0

Peter Luschny, Jun 06 2016

Cf. A000012 (m=1), A002596 (m=2), A273192, A273194.

Blist := proc(m, size) local H, S;
H := m -> hypergeom([seq(k/m, k=1..m-1)], [], (x/m)^m);
S := m -> series(1/H(m), x, (m+1)*size);
seq((-1)^n*numer(coeff(S(m), x, m*n)), n=0..size) end:
A273193_list := size -> Blist(3, size);

A364660 Numerators of coefficients in expansion of (1 + x)^(1/4).

Original entry on oeis.org

1, 1, -3, 7, -77, 231, -1463, 4807, -129789, 447051, -3129357, 11094993, -159028233, 574948227, -4188908511, 15359331207, -906200541213, 3358272593907, -25000473754641, 93422822977869, -1401342344668035, 5271716439465465, -39777496770512145, 150462705175415505, -4564035390320936985

Offset: 0

Ilya Gutkovskiy, Aug 01 2023

			(1 + x)^(1/4) = 1 + x/4 - 3*x^2/32 + 7*x^3/128 - 77*x^4/2048 + 231*x^5/8192 - 1463*x^6/65536 + ...
Coefficients are 1, 1/4, -3/32, 7/128, -77/2048, 231/8192, -1463/65536, ...

Denominators are A088802, A123854.

Cf. A002596, A004130, A008545, A067622, A364661.

nmax = 24; CoefficientList[Series[(1 + x)^(1/4), {x, 0, nmax}], x] // Numerator
Table[Binomial[1/4, n], {n, 0, 24}] // Numerator

my(x='x+O('x^30)); apply(numerator, Vec((1 + x)^(1/4))) \\ Michel Marcus, Aug 02 2023

A364661 Numerators of coefficients in expansion of (1 + x)^(3/4).

Original entry on oeis.org

1, 3, -3, 5, -45, 117, -663, 1989, -49725, 160225, -1057485, 3556995, -48612265, 168273225, -1177912575, 4161957765, -237231592605, 851242773465, -6147864475025, 22326455198775, -325966245902115, 1195209568307755, -8801088639357105, 32525762362841475, -964930950097630425

Offset: 0

Ilya Gutkovskiy, Aug 01 2023

			(1 + x)^(3/4) = 1 + 3*x/4 - 3*x^2/32 + 5*x^3/128 - 45*x^4/2048 + 117*x^5/8192 - 663*x^6/65536 + ...
Coefficients are 1, 3/4, -3/32, 5/128, -45/2048, 117/8192, -663/65536, ...

Denominators are A088802, A123854.

Cf. A002596, A067002, A364658, A364660.

nmax = 24; CoefficientList[Series[(1 + x)^(3/4), {x, 0, nmax}], x] // Numerator
Table[Binomial[3/4, n], {n, 0, 24}] // Numerator

my(x='x+O('x^30)); apply(numerator, Vec((1 + x)^(3/4))) \\ Michel Marcus, Aug 02 2023

A364658 Numerators of coefficients in expansion of (1 + x)^(2/3).

Original entry on oeis.org

1, 2, -1, 4, -7, 14, -91, 208, -494, 10868, -27170, 69160, -535990, 1401820, -3704810, 29638480, -79653415, 215532770, -5280552865, 14452039420, -39743108405, 329300041070, -913059204785, 2540686482880, -21278249294120, 59579098023536, -167279775219928, 12713262916714528

Offset: 0

Ilya Gutkovskiy, Aug 01 2023

			(1 + x)^(2/3) = 1 + 2*x/3 - x^2/9 + 4*x^3/81 - 7*x^4/243 + 14*x^5/729 - 91*x^6/6561 + ...
Coefficients are 1, 2/3, -1/9, 4/81, -7/243, 14/729, -91/6561, ...

Denominators are A067623.

Cf. A002596, A067622, A127974, A161200, A364661.

nmax = 27; CoefficientList[Series[(1 + x)^(2/3), {x, 0, nmax}], x] // Numerator
Table[Binomial[2/3, n], {n, 0, 27}] // Numerator

my(x='x+O('x^30)); apply(numerator, Vec((1 + x)^(2/3))) \\ Michel Marcus, Aug 02 2023

A381059 Array read by ascending antidiagonals: A(n,k) = numerator(binomial(n-1/2,k)) with k >=0.

Original entry on oeis.org

1, 1, -1, 1, 1, 3, 1, 3, -1, -5, 1, 5, 3, 1, 35, 1, 7, 15, -1, -5, -63, 1, 9, 35, 5, 3, 7, 231, 1, 11, 63, 35, -5, -3, -21, -429, 1, 13, 99, 105, 35, 3, 7, 33, 6435, 1, 15, 143, 231, 315, -7, -5, -9, -429, -12155, 1, 17, 195, 429, 1155, 63, 7, 5, 99, 715, 46189

Offset: 0

Stefano Spezia, Feb 12 2025

Numerators of the binomial coefficients for half-integers. The denominators are given by the absolute values of A173755.

			The array of the binomial coefficients for half-integers begins as:
  1, -1/2,  3/8,  -5/16,   35/128, -63/256, ...
  1,  1/2, -1/8,   1/16,   -5/128,   7/256, ...
  1,  3/2,  3/8,  -1/16,    3/128,  -3/256, ...
  1,  5/2, 15/8,   5/16,   -5/128,   3/256, ...
  1,  7/2, 35/8,  35/16,   35/128,  -7/256, ...
  1,  9/2, 63/8, 105/16,  315/128,  63/256, ...
  1, 11/2, 99/8, 231/16, 1155/128, 693/256, ...
  ...

Stefano Spezia, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals of the array)
Gábor Hegedüs, Sho Suda, and Ziqing Xiang, Codes with symmetric distances, arXiv:2501.11461 [math.CO], 2025. See p. 10.

Cf. A000466, A161200, A161202, A162540, A173755.
Columns k=0..1 give A000012, A060747.
Row n=1 gives A002596.
Main diagonal gives A001790.

A[n_,k_]:=Numerator[Binomial[n-1/2,k]]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=Numerator[(2n-1)!!/((2(n-k)-1)!!2^k k!)]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

A(n,k) = numerator((2*n - 1)!!/((2*(n - k) - 1)!!*2^k*k!)).
A(n,2) = A000466(n-1) for n > 0.
A(n,3) = A162540(n-3) for n > 3.
A(0,n) = (-1)^n*A001790(n).
abs(A(2,n)) = abs(A161200(n)).
abs(A(3,n)) = abs(A161202(n)).

A002461 Coefficients of Legendre polynomials.

Original entry on oeis.org

1, 3, 20, 35, 126, 231, 3432, 6435, 24310, 46189, 352716, 676039, 2600150, 5014575, 155117520, 300540195, 1166803110, 2268783825, 17672631900, 34461632205, 134564468610, 263012370465, 4116715363800, 8061900920775, 31602651609438, 61989816618513, 486734856412028

Offset: 2

N. J. A. Sloane

nonn

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 362.
G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

a(n) = if (n==2, 1, n*abs(numerator((1/(1-2*n))*binomial(2*n,n)/(4^n)))); \\ Michel Marcus, Feb 02 2015

1 if n=2, n * |A002596(n)| else. - Ralf Stephan, Sep 01 2003

More terms from Michel Marcus, Feb 02 2015

A186642 Decimal expansion of the "squircle" perimeter.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 25 2011

This squircle constant can also be computed as a series in terms of incomplete beta function with coefficients from sequences A002596 and A120777:
  a(n) = (-1)^(n+1) numerator((2n-3)!!/n!) ( sequence A002596);
  b(n) = denominator(binomial(2n+2, n+1)/2^(2n+1)) ( sequence A120777).
  Generic term:
  u(n) = (a(n)/b(n-1))*beta(1/2, (6n+1)/4, 1-(3/2)*n).
  Here is the series computed up to 5 terms:
  4*2^(3/4) + sum(u(n), {n, 1, 5}) =
  4*2^(3/4) + beta(1/2, 7/4, -1/2) - (1/4)*beta(1/2, 13/4, -2) + (1/8)* beta(1/2, 19/4, -7/2) - (5/64)*beta(1/2, 25/4, -5) + (7/128)*beta(1/2, 31/4, -13/2).
  It evaluates to 7.018901897260651...
  Numeric check with 10000 terms:
  4*2^(3/4) + sum(u(n), {n, 1, 10000}) = 7.017697943556135...

			7.01769794356404...

Eric Weisstein's World of Mathematics, Squircle

Cf. A175576 (unit squircle area).

First @ RealDigits[N[2*Integrate[Sqrt[1 + x^(3/2)/(1 - x)^(3/2)]/x^(3/4), {x, 0, 1/2}], 100]]
(* This other series formula gives 100 correct digits: *)
First @ RealDigits[1/Sqrt[Pi]*NSum[(-1)^(n+1)*Gamma[n - 1/2]*Beta[1/2, (6n + 1)/4, 1 - (3/2)n] / n!, {n, 0, Infinity},WorkingPrecision -> 100, Method -> "AlternatingSigns"], 10, 100]

-((3^(1/4) MeijerG[{{1/3, 2/3, 5/6, 1, 4/3}, {}}, {{1/12, 5/12, 7/12, 3/4, 13/12}, {}}, 1])/(16 Sqrt[2] Pi^(7/2) Gamma[5/4])). - Eric W. Weisstein, Oct 25 2011

A364713 a(n) is the numerator of coefficient of x^n in expansion of (1 + x)^(1/n).

Original entry on oeis.org

1, -1, 5, -77, 399, -124729, 81549, -23960365, 283583443, -478398640447, 19740912828, -11911591259019739, 18262332208600, -4514446693068714225, 142267808222130386191, -1912831808055538077885, 39773048560156838355, -43025628065750129034887540875, 86435429204640847578555

Offset: 1

Ilya Gutkovskiy, Aug 04 2023

			1, -1/8, 5/81, -77/2048, 399/15625, -124729/6718464, 81549/5764801, ...

Denominators are A145921.

Cf. A002596, A067622, A344745, A364660.

Table[SeriesCoefficient[(1 + x)^(1/n), {x, 0, n}], {n, 1, 21}] // Numerator
Table[Binomial[1/n, n], {n, 1, 21}] // Numerator

a(n) = numerator(binomial(1/n, n)); \\ Michel Marcus, Aug 05 2023

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A001795 Coefficients of Legendre polynomials.

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A224270 Absolute values of the numerators of the third column of ( 0 followed by (interleave 0 , A001803(n))/A060818(n) ) and its successive differences.

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A273193 Numerators of the nonzero coefficients in the expansion of 1/hypergeom([Seq_{k=1..m-1} k/m], [], (x/m)^m) for m = 3.

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A364660 Numerators of coefficients in expansion of (1 + x)^(1/4).

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A364661 Numerators of coefficients in expansion of (1 + x)^(3/4).

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A364658 Numerators of coefficients in expansion of (1 + x)^(2/3).

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A381059 Array read by ascending antidiagonals: A(n,k) = numerator(binomial(n-1/2,k)) with k >=0.

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A002461 Coefficients of Legendre polynomials.

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