Karol A. Penson - cp's OEIS frontend

A386517 a(n) = 60binomial(3n,n)/(n+2).

30, 60, 225, 1008, 4950, 25740, 139230, 775200, 4412826, 25564500, 150225075, 893246400, 5364333000, 32489701776, 198225859050, 1217179149120, 7516163045490, 46645481326500, 290779045735815, 1819954198062000, 11432303197651350, 72050701049642700, 455457919954401000

Offset: 0

Karol A. Penson, Aug 25 2025

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A005809.

A386517[n_] := 60*Binomial[3*n, n]/(n + 2); Array[A386517, 25, 0] (* Paolo Xausa, Aug 28 2025 *)

a(n) = 60*binomial(3*n,n)/(n+2);
vector(23,n,a(n-1)) \\ Joerg Arndt, Aug 26 2025

a(n) = 60*A005809(n)/(n+2).
O.g.f.: (-i*(x + 1)*x*sqrt(3)*t - 4*sqrt(x) + 15*x^(3/2) + 81*x^(5/2))*(4*t - 12*i*sqrt(3)*sqrt(x))^(1/3) + (i*(x + 1)*x*sqrt(3)*t - 4*sqrt(x) + 15*x^(3/2) + 81*x^(5/2))*(4*t + 12*i*sqrt(3)*sqrt(x))^(1/3) + 8*t*sqrt(x))/(4*t*x^(5/2)), for t = sqrt(4-27*x), and i = sqrt(-1) the imaginary unit.
O.g.f.: 30*hypergeom([1/3, 2/3, 2], [1/2, 3], 27*x/4), that, denoted by h(x),satisfies
  -270 - 540*x + 675*x^2 + 1728*x^3 + 9*(1 + 10*x^2 + 20*x^3)*h(x) - 6*x^2*h(x)^2 + x^4*h(x)^3 = 0.
E.g.f.: 30*hypergeom([1/3, 2/3, 2], [1/2, 1, 3], 27*x/4).
a(n) = Integral_{x=0..27/4} x^n*W(x)*dx, where, for S = sqrt(27 - 4*x),
W(x) = ((-6^(1/3)*(9 + sqrt(3)*S)^(2/3)*(sqrt(3) - 3*S) + 6*3^(1/6)*(1 + sqrt(3)*S)*x^(1/3) + 6^(1/3)*(9 + sqrt(3)*S)^(2/3)*(-sqrt(3) + S)*x + 2*3^(1/6)*(3 + sqrt(3)*S)*x^(4/3))*2^(1/3))/(8*Pi*(9 + sqrt(3)*S)^(1/3)*x^(2/3)).
W(x) is a positive function on x = (0,27/4), is singular at x=0, and tends to zero at x = 27/4. Thus a(n) is a positive definite sequence. This representation is unique as W(x) is the solution of the Hausdorff moment problem.

A384957 Expansion of g.f.: exp(Sum_{n>=1} A295433(n)*x^n/n).

Original entry on oeis.org

1, 990, 2206149, 6450139410, 21553605027306, 77957908218716988, 297118041166459732781, 1175248212459867447863562, 4779368947089383238327733950, 19858241947988743766121587718308, 83936671517628352407663509802203682, 359778601391313651280693986124971038388, 1560159110515342136997114532804454280500084

Offset: 0

Karol A. Penson, Jun 13 2025

nonn

Cf. A295432, A255881, A243953, A166990, A229451, A383796.

seq(n)=Vec(exp(sum(n=1, n, (12*n)!*n!*x^n/((8*n)!*(3*n)!*(2*n)!)/n, O(x*x^n)) )) \\ Andrew Howroyd, Jun 13 2025

G.f.: exp(Sum_{n>=1} (12*n)!*n!*x^n/((8*n)!*(3*n)!*(2*n)!)/n).

A384668 a(n) = 12 * (5n+2)! / ((3n+1)! * (2*n+2)!).

Original entry on oeis.org

12, 105, 1584, 29172, 596904, 13037895, 297748800, 7023149820, 169774618104, 4183919862474, 104722807600320, 2654939113240050, 68033328627480804, 1759318006963275528, 45853277234783179392, 1203249937243079847660, 31764232607604306053400, 842982010030680328418706

Offset: 0

Karol A. Penson, Jun 06 2025

nonn

Paolo Xausa, Table of n, a(n) for n = 0..650

Cf. A384585, A002293, A000260, A001450.

A384668[n_] := 12*(5*n + 2)!/((3*n + 1)!*(2*n + 2)!);
Array[A384668, 20, 0] (* Paolo Xausa, Jun 12 2025 *)

O.g.f.: 12*hypergeom([3/5, 4/5, 1, 6/5, 7/5], [2/3, 4/3, 3/2, 2], (3125*x)/108).
E.g.f.: 12*hypergeom([3/5, 4/5, 6/5, 7/5], [2/3, 4/3, 3/2, 2], (3125*x)/108).
O.g.f. denoted by h(x), satisfies the algebraic equation of order 10:
1889568 - 6141096*x + 10628820*x^2 - 59049*x^3 + (-2834352*x^3 + 4861701*x^2 - 2834352*x - 157464)*h(x) + 13122*x*(14*x^3 - 77*x^2 + 124*x + 30)*h(x)^2 - 4374*x^2*(14*x^2 + 94*x + 99)*h(x)^3 +  729*x^3*(50*x^2 + 32*x + 377)*h(x)^4 - 243*x^4*(11*x^2 - 40*x + 456)*h(x)^5 - 243*x^5*(8*x - 121)*h(x)^6 + 54*x^6*(2*x - 95)*h(x)^7 + 567*x^7*h(x)^8 - 36*x^8*h(x)^9 + x^9*h(x)^10 = 0.
a(n) = Integral_{x=0..3125/108} x^n*W(x)*dx, where W(x) = W1(x)+W2(x)+W3(x)+W4(x), with
  W1(x) = (3*sqrt(5)*csc(Pi/5)*sin(Pi/10)*hypergeom([-2/5, 1/10, 4/15, 14/15], [1/5, 2/5, 4/5], (108*x)/3125))/(2*Pi*x^(2/5)),
  W2(x) = (6*sqrt(5)*csc((2*Pi)/5)*sin((3*Pi)/10)*hypergeom([-1/5, 3/10, 7/15, 17/15], [2/5, 3/5, 6/5], (108*x)/3125))/(5*Pi*x^(1/5)),
  W3(x) = -(24*sqrt(5)*csc((2*Pi)/5)*sin((3*Pi)/10)*x^(1/5)*hypergeom([1/5, 7/10, 13/15, 23/15], [4/5, 7/5, 8/5], (108*x)/3125))/(125*Pi), and
  W4(x) = -(33*sqrt(5)*csc(Pi/5)*sin(Pi/10)*x^(2/5)*hypergeom([2/5, 9/10, 16/15, 26/15], [6/5, 8/5, 9/5], (108*x)/3125))/(1250*Pi).
This integral representation is unique as it is the solution of the Hausdorff power moment of the function W(x). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0 and for x > 0 is monotonically decreasing to zero at x = 3125/108. Therefore a(n) is a positive definite sequence.

A383796 Expansion of g.f.: exp(Sum_{n>=1} A295432(n)*x^n/n).

Original entry on oeis.org

1, 462, 396453, 425295010, 511915968714, 661059663660060, 895093835464198893, 1254056426977089876570, 1802794259810040618367902, 2644298823194748929633091780, 3941742074897786728895080586082, 5954164159064906497558129244865108, 9094122817144126105637193154022530612

Offset: 0

Karol A. Penson, Jun 11 2025

nonn

Cf. A295432, A255881, A243953, A166990, A229451.

seq(n)=Vec(exp(sum(n=1, n, (12*n)!*(3*n)!*(2*n)!*x^n/(n*((6*n)!)^2*(4*n)!*n!), O(x*x^n)))) \\ Andrew Howroyd, Jun 11 2025

G.f.: exp(Sum_{n>=1} (12*n)!*(3*n)!*(2*n)!*x^n/(n*((6*n)!)^2*(4*n)!*n!)).

A384585 a(n) = 6 * (4n)! / ((n+1)! (3*n+1)!).

Original entry on oeis.org

6, 3, 8, 33, 168, 969, 6072, 40365, 280488, 2017356, 14914848, 112784399, 869046168, 6803716710, 53997506640, 433647466245, 3518801467560, 28815074239908, 237887596740192, 1978246301709540, 16558857808956320, 139428557033056785, 1180350813375438840, 10041660963789578955

Offset: 0

Karol A. Penson, Jun 04 2025

nonn

Since a(1) < a(0) the sequence is not growing monotonically with n.

Cf. A002293, A000260.

a:= proc(n) option remember; `if`(n<2, 6-3*n,
      8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)/(3*(3*n-1)*(3*n+1)*(n+1)))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 04 2025

a[n_]:=6*(4*n)!/((n+1)!*(3*n+1)!); Array[a,24,0] (* Stefano Spezia, Jun 04 2025 *)

O.g.f.: 6*hypergeom([1/4, 1/2, 3/4, 1], [2/3, 4/3, 2], (256*z)/27).
E.g.f.: 6*hypergeom([1/4, 1/2, 3/4], [2/3, 4/3, 2], (256*z)/27).
O.g.f. = h(z) satisfies algebraic equation of order 4: -6 - 39*z + 4096*z^2 + (1 - 12*z - 768*z^2)*h(z) - 3*z*(2*z - 1)*h(z)^2 + 3*z^2*h(z)^3 + z^3*h(z)^4 = 0.
a(n) = Integral_{x=0..256/27} x^n*W(x)*dx, where W(x) = W1(x)+W2(x)+W3(x), with
  W1(x) = 4*sqrt(2)*hypergeom([-3/4, -1/12, 7/12], [1/2, 3/4], (27*x)/256)/(Pi*x^(3/4)),
  W2(x) = -3*hypergeom([-1/2, 1/6, 5/6], [3/4, 5/4], (27*x)/256)/(Pi*sqrt(x)), and
  W3(x) = -3*sqrt(2)*hypergeom([-1/4, 5/12, 13/12], [5/4, 3/2], (27*x)/256)/(8*Pi*x^(1/4)).
This integral representation is unique as it is the solution of the Hausdorff power moment problem of the function W(x). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0 and for x > 0 is monotonically decreasing to zero at x = 256/27. Therefore a(n) is a positive definite sequence.

A384542 Expansion of g.f. sinh(7arctanh(14sqrt(x)))/(98*sqrt(x)).

Original entry on oeis.org

1, 1666, 1090054, 485318932, 176760328262, 56963958713340, 16909346921973660, 4732136004374122344, 1266899066122354262598, 327667319343098397330668, 82435716917761454374571444, 20275150472587631020453400984, 4893425028040341625551135687452, 1162305136998381407493307772297560

Offset: 0

Karol A. Penson, Jun 02 2025

nonn

Cf. A383928, A384335, A285043, A285044, A285045, A285046, A384417.

a[n_]:=49^n*(105 + 464*n + 704*n^2 + 512*n^3)*(2*n)!/(105*(n!)^2); Array[a,14,0] (* Stefano Spezia, Jun 02 2025 *)

a(n) = 49^n*(105 + 464*n + 704*n^2 + 512*n^3)*(2*n)!/(105*(n!)^2).
O.g.f.: (1 + 980*x + 115248*x^2 + 1075648*x^3)/(-196*x + 1)^(7/2).
E.g.f.: exp(98*x)*(BesselI(0, 98*x)*(275365888*x^3 + 5444096*x^2 + 23520*x + 15) + 224*x*BesselI(1, 98*x)*(1229312*x^2 + 18032*x + 29))/15.

A384417 Expansion of g.f. cosh(9arctanh(8sqrt(x))).

Original entry on oeis.org

1, 2592, 1230336, 294469632, 49690312704, 6822215811072, 818458027622400, 89312567167549440, 9086229152658358272, 875874088323041460224, 80899222450192930308096, 7217466034064795168145408, 625687045828728598806134784, 52946875811413468120885493760, 4389120887020725640048536453120

Offset: 0

Karol A. Penson, May 28 2025

nonn

Cf. A383928, A384335, A285043, A285044, A285045, A285046.

seq(coeff(series((1 + 2304*x + 516096*x^2 + 22020096*x^3 + 150994944*x^4)/(-64*x + 1)^(9/2), x, 15), x, k), k=0..14);

CoefficientList[Series[Cosh[9*ArcTanh[8*Sqrt[x]]],{x,0,14}],x] (* Stefano Spezia, May 29 2025 *)

a(n) = 16^n*(105 + 400*n + 3392*n^2 + 512*n^3 + 4096*n^4)*(2*n)!/(105*(n!)^2).
O.g.f.: (1 + 2304*x + 516096*x^2 + 22020096*x^3 + 150994944*x^4)/(-64*x + 1)^(9/2).
E.g.f.: exp(32*x)*((105 + 512*x*(269 + 256*x*(73 + 512*x)))*BesselI(0, 32*x) + 512*x*(25 + 256*x*(65 + 512*x))*BesselI(1, 32*x))/105 + (131072*x*hypergeom([3/2, 2, 2], [1, 1, 1], 64*x))/105.

A384335 Expansion of g.f.: cosh(7arctanh(6sqrt(x))).

Original entry on oeis.org

1, 882, 150822, 14431284, 1052738694, 65805858972, 3724625506140, 196735568051880, 9876433300259526, 476865669055691916, 22326189769485093492, 1019514155600973935448, 45604820017276687744668, 2004918589790139365901720, 86848896758228990302070520, 3714470212008822424691576400

Offset: 0

Karol A. Penson, May 26 2025

nonn

Cf. A383928.

CoefficientList[Series[Cosh[7*ArcTanh[6*Sqrt[x]]],{x,0,15}],x] (* Stefano Spezia, May 26 2025 *)

a(n) = 9^n*(512*n^3 + 64*n^2 + 144*n + 15)*(2*n)!/(15*(n!)^2).
G.f.: (1 + 756*x + 45360*x^2 + 326592*x^3)/(-36*x + 1)^(7/2).
E.g.f.: (1/5)*exp(18*x)*((5 + 4320*x +  290304*x^2 + 3981312*x^3)*BesselI(0,18*x)+((864*x + 179712*x^2 + 3981312*x^3)*BesselI(1,18*x))).

A383874 a(n) = (3n+1)!(3n)!/((2n)!*((n+1)!)^2).

Original entry on oeis.org

1, 18, 4200, 3175200, 5137292160, 14544244915200, 64008493310361600, 405192226643043840000, 3493057136053143859200000, 39378260464472988708249600000, 562659674639968187756457984000000, 9940535265182157971578474463232000000, 212816707229761791940688046273331200000000

Offset: 0

Karol A. Penson, May 22 2025

nonn

Paolo Xausa, Table of n, a(n) for n = 0..150

Cf. A166149, A166771, A166494, A166384, A166750, A271049, A064352.

A383874[n_] := (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2);
Array[A383874, 15, 0] (* Paolo Xausa, May 26 2025 *)

a(n) = (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2); \\ Michel Marcus, May 22 2025

O.g.f.: hypergeom([1/3, 2/3, 2/3, 1, 1, 4/3], [1/2, 2, 2], (729*x)/4).
E.g.f.: hypergeom([1/3, 2/3, 2/3, 1, 1, 4/3], [1/2, 2, 2, 1], (729*x)/4).
a(n) = Integral_{x>=0} x^n*W(x)*dx, n>=0, with W(x) = MeijerG([[],[-1/2,1,1]],[[0,-1/3,-1/3,1/3,-2/3],[]],4*x/729)/(81*Pi^(3/2)), where MeijerG is the Meijer G - function. Apparently W(x) cannot be represented by any other simpler functions. W(x) is a positive function on (0,oo), is singular at x = 0 and goes monotonically to zero as x -> oo. Thus a(n) is a positive definite sequence.
  W(x) is the solution of the Stieltjes moment problem and it may be non-unique.
a(n) ~ 3^(6*n+2) * n^(2*n - 3/2) / (sqrt(Pi) * 2^(2*n+1) * exp(2*n)). - Vaclav Kotesovec, May 24 2025

A383928 Expansion of g.f. cosh(9arctanh(4sqrt(x))).

Original entry on oeis.org

1, 648, 76896, 4601088, 194102784, 6662320128, 199818854400, 5451206492160, 138644854013952, 3341194489757696, 77151510667984896, 1720777996555517952, 37293854107184922624, 788969931176505507840, 16350749459194860011520, 332885987884833366343680, 6673058165121160335851520

Offset: 0

Karol A. Penson, May 15 2025

Cf. A285043, A285044, A285045, A285046.

seq(coeff(series((589824*x^4 + 344064*x^3 + 32256*x^2 + 576*x + 1)/(-16*x + 1)^(9/2), x, 17), x, k), k=0..16);

a(n) = 4^n*(105 + 400*n + 3392*n^2 + 512*n^3 + 4096*n^4)*(2*n)!/(105*(n!)^2).
O.g.f.: (1 + 576*x + 32256*x^2 + 344064*x^3 + 589824*x^4)/(-16*x + 1)^(9/2).
E.g.f.: 134217728*((x^4 + 41/128*x^3 + 425/16384*x^2 + 525/1048576*x + 105/134217728)*BesselI(0, 8*x) + x*BesselI(1, 8*x)*(x^3 + 33/128*x^2 + 193/16384*x + 25/1048576))*exp(8*x)/105.

cp's OEIS Frontend

User: Karol A. Penson

A386517 a(n) = 60*binomial(3*n,n)/(n+2).

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A384957 Expansion of g.f.: exp(Sum_{n>=1} A295433(n)*x^n/n).

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A384668 a(n) = 12 * (5*n+2)! / ((3*n+1)! * (2*n+2)!).

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A383796 Expansion of g.f.: exp(Sum_{n>=1} A295432(n)*x^n/n).

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A384585 a(n) = 6 * (4*n)! / ((n+1)! * (3*n+1)!).

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A384542 Expansion of g.f. sinh(7*arctanh(14*sqrt(x)))/(98*sqrt(x)).

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A384417 Expansion of g.f. cosh(9*arctanh(8*sqrt(x))).

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A384335 Expansion of g.f.: cosh(7*arctanh(6*sqrt(x))).

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A383874 a(n) = (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2).

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A386517 a(n) = 60binomial(3n,n)/(n+2).

A384668 a(n) = 12 * (5n+2)! / ((3n+1)! * (2*n+2)!).

A384585 a(n) = 6 * (4n)! / ((n+1)! (3*n+1)!).

A384542 Expansion of g.f. sinh(7arctanh(14sqrt(x)))/(98*sqrt(x)).

A384417 Expansion of g.f. cosh(9arctanh(8sqrt(x))).

A384335 Expansion of g.f.: cosh(7arctanh(6sqrt(x))).

A383874 a(n) = (3n+1)!(3n)!/((2n)!*((n+1)!)^2).

A383928 Expansion of g.f. cosh(9arctanh(4sqrt(x))).