A007272 -id:A007272 - cp's OEIS frontend

A156126 Sequence related to Hankel transform of super-ballot numbers.

1, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, 2925, 3654, 4495, 5456, 6545, 7770, 9139, 10660, 12341, 14190, 16215, 18424, 20825, 23426, 26235, 29260, 32509, 35990, 39711, 43680

Offset: 0

Paul Barry, Feb 04 2009

Hankel transform of A007272 is 10,35,84,... with g.f. (10-5x+4x^2-x^3)/(1-x)^4.

Hankel transform of A156125 is 10^(n^2-1+0^n)*A156126(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

I:=[1, 35, 84, 165, 286]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 30 2012

CoefficientList[Series[(1+31x-50x^2+35x^3-9x^4)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 30 2012 *)
LinearRecurrence[{4,-6,4,-1},{1,35,84,165,286},40] (* Harvey P. Dale, Mar 25 2022 *)

G.f.: (1+31x-50x^2+35x^3-9x^4)/(1-x)^4.

a(n) = (2*n+5)*(2*n+3)*(n+2)/3, n>0. - R. J. Mathar, Oct 13 2011

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}]]

A361040 a(n) = 420(3n)!/(n!(2n + 3)!).

Original entry on oeis.org

70, 21, 30, 70, 210, 735, 2856, 11970, 53130, 246675, 1187550, 5890248, 29954680, 155602020, 823184880, 4424618730, 24116031162, 133072694475, 742405558650, 4182821562150, 23776769743650, 136248095712855, 786482994679200

Offset: 0

Peter Bala, Mar 04 2023

The Catalan numbers A000108 are defined by the formula Catalan(n) = (2*n)!/(n!*(n+1)!). Gessel (1992) considered generalized Catalan numbers defined by Catalan(r,n) = J(r)*(2*n)!/(n!*(n+r+1)!), where J(r) = (2^r)*Product_{j = 0..r} (2*j + 1) is chosen so that these numbers are always integers. Gessel's generalized Catalan numbers are particular cases of super ballot numbers. See A135573 for a table of these generalized Catalan numbers.

For r = 0,1,2,..., it appears that there is an integer C(r) such the sequence {C(r)*(3*n)!/(n!*(2*n + r)!) : n >= 0} is integral. This is the case r = 3. For other cases see A005809 (r = 0, C(0) = 1), A001764 (r = 1, C(1) = 1), A000139 (r = 2, C(2) = 4) and A361041 (r = 4, C(4) = 1680).

Ira M. Gessel, Super ballot numbers, J. Symbolic Comp., 14 (1992), 179-194.

Cf. A000139, A001764, A005809, A007272, A135573, A361038, A361041.

seq( 420*(3*n)!/(n!*(2*n + 3)!), n = 0..20)

a(n) = 70*binomial(3*n,2*n) - 189*binomial(3*n,2*n+1) + 114*binomial(3*n,2*n+2) -32*binomial(3*n,2*n+3). Thus a(n) is an integer.
P-recursive: 2*(n + 1)*(2*n + 3)*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 70.
a(n) ~ (27/4)^n * 105*sqrt(3/(16*Pi))/n^(7/2).
The o.g.f. A(x) satisfies the differential equation
x^2*(4 - 27*x^4)*A''(x) + 2*x*(7 - 27*x)*A'(x) + (6 - 6*x)*A(x) - 420 = 0, with A(0) = 70 and A'(0) = 21.

A387249 a(n) = 10/(n + 1) * Catalan(3*n).

Original entry on oeis.org

10, 25, 440, 12155, 416024, 16158075, 682341000, 30582833775, 1433226830360, 69533550916004, 3468169547356640, 176946775343535925, 9199844912200348840, 486018122664268428850, 26029619941269629306160, 1410698658798280045783575, 77251704848334920869407000, 4269325372507953547350453420

Offset: 0

Peter Bala, Aug 24 2025

Compare with Catalan(n) = 1/(n + 1) * binomial(2*n, n).

For r >= 2, there is a constant K_r such that K_r/(n + 1) * Catalan(r*n) is integral for all n.

Paolo Xausa, Table of n, a(n) for n = 0..500

Cf. A000108, A000984, A007272, A066802, A387248, A387250.

seq( 10/((n+1)*(3*n+1)) * binomial(6*n, 3*n), n = 0..20);

A387249[n_] := 10*CatalanNumber[3*n]/(n + 1); Array[A387249, 20, 0] (* Paolo Xausa, Sep 02 2025 *)

a(n) = 10/((n + 1)*(3*n + 1)) * binomial(6*n, 3*n).
a(n) = (3*n + 2)/2 * (16*Catalan(3*n) - 8*Catalan(3*n+1) + Catalan(3n+2)) (shows a(n) to be an integer since Catalan(n) is odd iff n = 2^k - 1 for some k, so Catalan(3*n+2) is always even).
a(n) = (3*n + 2)/2 * A007272(3*n).
a(n) = 8*(2*n - 1)*(6*n - 1)*(6*n - 5)/((n + 1)*(3*n + 1)*(3*n - 1)) * a(n-1) with a(0) = 10.
a(n) ~ 10/(sqrt(27*Pi)) * 64^n/n^(5/2).
E.g.f.: 10*hypergeom([1/6, 1/2, 5/6], [2/3, 4/3, 2], 64*x). - Stefano Spezia, Aug 27 2025

A156125 a(n)=10^n*C(2n,n)/C(n+3,3).

Original entry on oeis.org

1, 5, 60, 1000, 20000, 450000, 11000000, 286000000, 7800000000, 221000000000, 6460000000000, 193800000000000, 5943200000000000, 185725000000000000, 5899500000000000000, 190095000000000000000, 6203100000000000000000

Offset: 0

Paul Barry, Feb 04 2009

Hankel transform is 10^(n^2-1+0^n)*A156126(n).

Table[10^n Binomial[2n,n]/Binomial[n+3,3],{n,0,20}] (* Harvey P. Dale, Mar 30 2022 *)

G.f.: F(1/2,1;4;40x);
a(n)=6*10^n*A000108(n)/((n+2)(n+3))=10^(n-1)*A007272(n).
(n+3)*a(n) +20*(1-2*n)*a(n-1)=0. - R. J. Mathar, Oct 25 2012

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]]

A218440 Hankel transform of A078818.

Original entry on oeis.org

10, 135, 1844, 25145, 342846, 4674655, 63738280, 869062689, 11849550290, 161566989191, 2202943686300, 30036834314425, 409548106582534, 5584132130887935, 76138873929651536, 1038143078887634945, 14154938162574828570, 193000635905606023879, 2631537137933532600580

Offset: 0

Arkadiusz Wesolowski, Oct 28 2012

Cf. A007272, A000447, A078818.

CoefficientList[Series[(10-5x+14x^2-x^3)/(1-14x+6x^2-14x^3+x^4),{x,0,30}],x] (* Harvey P. Dale, Dec 09 2018 *)

G.f.: (10 - 5*x + 14*x^2 - x^3)/(1 - 14*x + 6*x^2 - 14*x^3 + x^4).

cp's OEIS Frontend

A156126 Sequence related to Hankel transform of super-ballot numbers.

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Mathematica

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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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A361040 a(n) = 420*(3*n)!/(n!*(2*n + 3)!).

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Maple

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A387249 a(n) = 10/(n + 1) * Catalan(3*n).

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Maple

Mathematica

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A156125 a(n)=10^n*C(2n,n)/C(n+3,3).

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Mathematica

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

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Mathematica

A218440 Hankel transform of A078818.

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Mathematica

Formula

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

A361040 a(n) = 420(3n)!/(n!(2n + 3)!).