A002421 -id:A002421 - cp's OEIS frontend

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

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)

A320827 G.f.: -sqrt(1 - 4x)(2x - 1)/(3x - 1).

Original entry on oeis.org

-1, 1, 1, 3, 11, 41, 151, 549, 1977, 7075, 25229, 89831, 319881, 1140523, 4075321, 14603243, 52501659, 189440937, 686181711, 2495243373, 9109701699, 33388293177, 122840931891, 453622854873, 1681057537359, 6250742452125, 23316503569983, 87236431248445

Offset: 0

Peter Luschny, Oct 23 2018

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A002421, A029759, A320825, A320826.

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt(1-4*x)*(1-2*x)/(3*x-1))); // G. C. Greubel, Oct 27 2018

ogf := x -> -sqrt(1 - 4*x)*(2*x - 1)/(3*x - 1);
ser := series(ogf(x), x, 30); seq(coeff(ser, x, k), k=0..27);
# By recurrence:
a := proc(n) option remember; if n <= 4 then return [-1,1,1,3,11][n+1] fi;
((-90+66*n-12*n^2)*a(n-2)+(30-34*n+7*n^2)*a(n-1))/((n-4)*n) end:
seq(a(n), n=0..27);

a[n_] := (-4)^n Binomial[3/2,n]((4/3)n - 2 + Hypergeometric2F1[1,-n, 5/2 - n, 3/4]); Table[a[n], {n, 0, 27}]
CoefficientList[Series[Sqrt[1-4*x]*(1-2*x)/(3*x-1), {x, 0, 40}], x] (* G. C. Greubel, Oct 27 2018 *)

x='x+O('x^40); Vec(sqrt(1-4*x)*(1-2*x)/(3*x-1)) \\ G. C. Greubel, Oct 27 2018

a(n) = (-4)^n*binomial(3/2, n)*((4/3)*n - 2 + hypergeom([1, -n], [5/2 - n], 3/4)).
D-finite with recurrence: a(n) = ((-90+66*n-12*n^2)*a(n-2) + (30-34*n+7*n^2)*a(n-1))/((n-4)*n) for n >= 5.
Expansion of -1/g.f. gives A029759.
a(n) = A320825(n) - A320826(n).

A184881 a(n) = A184879(2n, n) - A184879(2n, n+1) where A184879(n, k) = Hypergeometric2F1(-2k, 2k-2*n, 1, -1) if 0<=k<=n.

Original entry on oeis.org

1, -3, 2, -3, 6, -14, 36, -99, 286, -858, 2652, -8398, 27132, -89148, 297160, -1002915, 3421710, -11785890, 40940460, -143291610, 504932340, -1790214660, 6382504440, -22870640910, 82334307276, -297670187844, 1080432533656, -3935861372604, 14386251913656

Offset: 0

Paul Barry, Jan 24 2011

sign

Hankel transform is A184882.

Signed version of A007054. - Philippe Deléham, Mar 19 2014

			a(0) = 1;
a(1) = 1 - 4*1 = -3;
a(2) = 4*1 - 2 = 2;
a(3) = 5 - 4*2 = -3;
a(4) = 4*5 - 14 = 6;
a(5) = 42 - 4*14 = -14;
a(6) = 4*42 - 132 = 36;
a(7) = 429 - 4*132 = -99;
a(8) = 4*429 - 1430 = 286, etc; with A000108 = 1,1,2,5,14,42,132,429,1430, ... - _Philippe Deléham_, Mar 19 2014
G.f. = 1 - 3*x + 2*x^2 - 3*x^3 + 6*x^4 - 14*x^5 + 36*x^6 - 99*x^7 + ... - _Michael Somos_, Mar 13 2023

J. W. Layman, The Hankel Transform and Some of Its Properties, J. Integer Sequences 4, No. 01.1.5, 2001
Fumitaka Yura, Hankel Determinant Solution for Elliptic Sequence, arXiv:1411.6972 [nlin.SI], (25-November-2014); see p. 7

Cf. A000108, A002421, A007054, A184879, A184882.

A184879 := proc(n,k) if k<0 or k >n then 0; else hypergeom([-2*k,2*k-2*n],[1],-1) ; simplify(%) ; end if; end proc:
A184881 := proc(n) A184879(2*n,n)-A184879(2*n,n+1) ; end proc:
seq(A184881(n),n=0..40) ; # R. J. Mathar, Feb 05 2011

h[n_, k_] := HypergeometricPFQ[{-2k, 2k - 2n}, {1}, -1];
a[0] = 1; a[n_] := h[2n, n] - h[2n, n + 1];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Nov 24 2017 *)

a(n) = 0^n + Sum_{k=0..2n} (C(2n,k)^2-C(2n+2,k)*C(2n-2,k))*(-1)^k.
G.f.: (8*x+1-sqrt(1+4*x)^3)/(2*x). - Philippe Deléham, Mar 19 2014
a(0) = 1, a(n) = (-1)^n*A007054(n-1) for n>0. - Philippe Deléham, Mar 19 2014
(n+1)*a(n) +2*(2*n-3)*a(n-1)=0. - R. J. Mathar, Nov 19 2014
a(n) = (-1)^n*A002421(n+1)/2 and 0 = a(n)*(+16*a(n+1) + 14*a(n+2)) + a(n+1)*(-6*a(n+1) + a(n+2)) for all n>0. - Michael Somos, Mar 13 2023

A320826 Expansion of x(1 - 4x)^(3/2)/(3*x - 1)^2.

Original entry on oeis.org

0, 1, 0, -3, -14, -51, -168, -521, -1542, -4365, -11740, -29439, -65670, -112273, -28344, 1018689, 6961550, 34606929, 151831044, 623095683, 2453975622, 9402575805, 35339538912, 130994480547, 480676041954, 1750847208621, 6343667488692, 22899720430251, 82466180250590

Offset: 0

Peter Luschny, Oct 22 2018

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A002421, A320825, A320827.

m:=30; R:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!(x*(1-4*x)^(3/2)/(1-3*x)^2)); // G. C. Greubel, Oct 27 2018

c := n -> (-4)^(n-1)*binomial(3/2, n-1):
h := n -> hypergeom([2, 1 - n], [7/2 - n], 3/4):
A320826 := n -> c(n)*h(n): seq(simplify(A320826(n)), n=0..28);

CoefficientList[Series[(x (1 -  4 x)^(3/2))/(3 x - 1)^2, {x, 0, 28}], x]

x='x+O('x^30); concat([0], Vec(x*(1-4*x)^(3/2)/(1-3*x)^2)) \\ G. C. Greubel, Oct 27 2018

a(n) = c(n)*h(n) where c(n) = Catalan(n)*(3*n*(n + 1))/(2*(2*n-5)*(2*n-3)*(2*n-1)) = (-4)^(n-1)*binomial(3/2, n-1) and h(n) = hypergeom([2, 1 - n], [7/2 - n], 3/4).

A320826(n) = A320825(n) - A320827(n).

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)

A085687 Expansion of g.f. 8/(1+sqrt(1-8*x))^3.

Original entry on oeis.org

1, 6, 36, 224, 1440, 9504, 64064, 439296, 3055104, 21498880, 152807424, 1095450624, 7911587840, 57511157760, 420459724800, 3089600348160, 22806128885760, 169033661153280, 1257467341701120, 9385880636620800, 70271680244613120, 527595313582571520

Offset: 0

N. J. A. Sloane, Jul 13 2003

nonn

a(n) is also the number of paths of length 2(n+1) in a binary tree between two vertices that are 2 steps apart. [David Koslicki (koslicki(AT)math.psu.edu), Nov 02 2010]

Cf. A002421, A000108, A000257.

CoefficientList[Series[8/(1 + Sqrt[1 - 8*x])^3, {x, 0, 21}], x] (* Amiram Eldar, Mar 24 2022 *)

a(n) = 6(n+1)*2^(n-2)*Cat(n+2)/(2n+3), where Cat(n)=A000108(n). - Ralf Stephan, Mar 11 2004
G.f.: c(2x)^3, where c(x) is the g.f. of A000108; a(n)=3(n+1)2^n*Cat(n+1)/(n+3); - Paul Barry, Dec 08 2004
a(n) = (n+1) * A000257(n+1). - F. Chapoton, Feb 26 2024
D-finite with recurrence: (n+3)*a(n) -2*(5*n+7)*a(n-1) +8*(2*n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2011
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 22/49 + 808*arcsin(1/(2*sqrt(2)))/(147*sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = 26/81 + 376*arcsinh(1/(2*sqrt(2)))/243. (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}]]

A232546 Expansion of (1 - 12*x)^(3/2) in powers of x.

Original entry on oeis.org

1, -18, 54, 108, 486, 2916, 20412, 157464, 1299078, 11258676, 101328084, 939587688, 8926083036, 86514343272, 852784240824, 8527842408240, 86344404383430, 883760374277460, 9132190534200420, 95167038198509640, 999253901084351220, 10563541240034570040

Offset: 0

Michael Somos, Nov 25 2013

sign

From Ralf Steiner, Apr 04 2017: (Start)
By analytic continuation to the entire complex plane there exist regularized values for divergent sums such as:
Sum_{k>=0} a(k)^2/16^k = 2F1(-3/2,-3/2,1,9).
Sum_{k>=0} a(k) / 6^k = -i. (End)

			G.f. = 1 - 18*x + 54*x^2 + 108*x^3 + 486*x^4 + 2916*x^5 + 20412*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..930
R. Steiner, Sums of OEIS-A232546, ResearchGate, 2017. - _Ralf Steiner_, Apr 04 2017

Cf. A000168, A002421, A115903.

a[ n_] := SeriesCoefficient[ (1 - 12 x)^(3/2), {x, 0, n}];
Table[9/Sqrt[Pi] 12^n Gamma[-1/2 + n]/Gamma[2 + n], {n, -1, 20}] (* Ralf Steiner, Apr 01 2017 *)
Flatten[{1, -18, Table[4*3^(n+1)*(2*n-4)!/((n-2)!*n!), {n, 2, 25}]}] (* Vaclav Kotesovec, Apr 02 2017 *)

{a(n) = if( n<0, 0, polcoeff( (1 - 12 * x + x * O(x^n))^(3/2), n))};

0 = a(n+2)*(a(n+1) - 42*a(n)) + 18*a(n+1)*(a(n+1) + 8*a(n)) for all n in Z.
a(n+2) = 54 * A000168(n). a(n) = 3^n * A002421(n). Convolution inverse of A115903.
a(n) = 6*(2*n-5)*a(n-1)/n. - R. J. Mathar, Nov 23 2014
G.f.: 1F0(-3/2;;12x). - R. J. Mathar, Aug 09 2015
For n>=2, a(n) = 4*3^(n+1)*(2*n-4)! / ((n-2)!*n!). - Vaclav Kotesovec, Apr 02 2017
Sum_{k>=0} a(k) / 12^k = 0. - Ralf Steiner, Apr 04 2017

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

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A320827 G.f.: -sqrt(1 - 4*x)*(2*x - 1)/(3*x - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A184881 a(n) = A184879(2*n, n) - A184879(2*n, n+1) where A184879(n, k) = Hypergeometric2F1(-2*k, 2*k-2*n, 1, -1) if 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A320826 Expansion of x*(1 - 4*x)^(3/2)/(3*x - 1)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A085687 Expansion of g.f. 8/(1+sqrt(1-8*x))^3.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

A232546 Expansion of (1 - 12*x)^(3/2) in powers of x.

Views

Author

Keywords

Comments

Examples

A320827 G.f.: -sqrt(1 - 4x)(2x - 1)/(3x - 1).

A184881 a(n) = A184879(2n, n) - A184879(2n, n+1) where A184879(n, k) = Hypergeometric2F1(-2k, 2k-2*n, 1, -1) if 0<=k<=n.

A320826 Expansion of x(1 - 4x)^(3/2)/(3*x - 1)^2.

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