A069723 -id:A069723 - cp's OEIS frontend

A258758 Triangle T(n,k) = C(n+k-1,k)C(2n-1,n-k).

1, 1, 1, 3, 6, 3, 10, 30, 30, 10, 35, 140, 210, 140, 35, 126, 630, 1260, 1260, 630, 126, 462, 2772, 6930, 9240, 6930, 2772, 462, 1716, 12012, 36036, 60060, 60060, 36036, 12012, 1716, 6435, 51480, 180180, 360360, 450450, 360360, 180180, 51480

Offset: 0

Vladimir Kruchinin, Jun 10 2015

Triangle T(n,k), read by rows, given by (1, 2, 1/2, 3/2, 2/3, 4/3, 3/4, 5/4, ...) DELTA (1, 2, 1/2, 3/2, 2/3, 4/3, 3/4, 5/4, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jun 19 2015

			[1]
[1,1]
[3,6,3]
[10,30,30,10]
[35,140,210,140,35]

Indranil Ghosh, Rows 0..100, flattened

Columns k=0-1 give: A088218, A002457(n-1) for n>0.

Cf. A069723 (row sums, with a shift).

[[Binomial(n+k-1,k)*Binomial(2*n-1,n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jun 12 2015

max = 10; s = (2*(x + y))/(-1 + 4*x + Sqrt[1 - 4*x - 4*y] + 4*y) + O[x]^(max+2) + O[y]^(max+2); t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]; Table[t[n - k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 10 2015, after Vladimir Kruchinin *)
Flatten[Table[Binomial[n+k-1,k] Binomial[2n-1,n-k], {n, 0, 9}, {k, 0, n}]] (* Indranil Ghosh, Mar 04 2017 *)

tabl(nn) = {for (n=0, nn, for(k=0, n, print1(binomial(n+k-1,k)*binomial(2*n-1,n-k),", "););print(););};
tabl(9); \\ Indranil Ghosh, Mar 04 2017

G.f.: A(x) = 1/(2 - C(x+y)), where C(x)=(1-sqrt(1-4*x))/(2*x) is g.f. of Catalan numbers (A000108).

It appears that T(n, k) = A088218(n)*binomial(n, k). - Michel Marcus, Jun 11 2015

A372506 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-2*x) )^n.

Original entry on oeis.org

1, 3, 23, 198, 1795, 16758, 159446, 1537308, 14967843, 146833830, 1449054178, 14369723316, 143072565454, 1429331585724, 14320668653580, 143838879376248, 1447883909314851, 14602334949928710, 147518977428892010, 1492559101878005700, 15121898521185194970

Offset: 0

Seiichi Manyama, May 04 2024

nonn

Cf. A001700, A069723, A085614, A255688, A372233.

A372506 := proc(n)
    add(binomial(n+k-1,k)*binomial(3*n-1,n-k),k=0..n) ;
end proc:
seq(A372506(n),n=0..80) ; # R. J. Mathar, Oct 24 2024

Table[SeriesCoefficient[1/((1 - x)*(1 - 2*x))^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 04 2024 *)
Table[Binomial[3*n - 1, n] * Hypergeometric2F1[-n, n, 2*n, -1], {n, 0, 20}] (* Vaclav Kotesovec, May 04 2024 *)

a(n, s=1, t=1, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t+u+1)*n-(s-1)*k-1, n-s*k));

a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(3*n-1,n-k).
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x) * (1-2*x) ).
a(n) ~ (1 + sqrt(3)) * 2^(n - 3/2) * 3^((3*n-1)/2) / sqrt(Pi*n). - Vaclav Kotesovec, May 04 2024
D-finite with recurrence 5*n*(n-1)*a(n) +18*(n-1)*(n-3)*a(n-1) +12*(-45*n^2+90*n-22)*a(n-2) -216*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Oct 24 2024

A128406 a(n) = (n+1)2^(n(n+1)).

Original entry on oeis.org

1, 8, 192, 16384, 5242880, 6442450944, 30786325577728, 576460752303423488, 42501298345826806923264, 12379400392853802748991242240, 14278816360970775978458864905355264, 65334214448820184984967924626899496599552, 1187470080331358621040493926581979953470445191168

Offset: 0

Paul Barry, Mar 01 2007

Hankel transform of A069723.

With offset 1, a(n) is the number of vertices with in-degree = 0 over all labeled digraphs (with self loops allowed) on n vertices. Equivalently, the number of elements in all labeled relations on an n-set that have no preimage. - Geoffrey Critzer, Aug 16 2016

Cf. A006125, A069723, A095340.

Table[n 2^(n - 1) 2^(n - 1)^2, {n, 1, 10}]
Table[(n+1)2^(n(n+1)),{n,0,20}] (* Harvey P. Dale, Jun 21 2021 *)

a(n) = A095340(n)*A006125(n+1).

A306625 Regular triangle T(n,k) = binomial(2n-2k,n-k)((n+1)/k)Sum_{k=0..floor((k-1)/2)} (-1)^kbinomial(2k,k)binomial(n+3k-2k,k-2k-1), read by rows.

Original entry on oeis.org

2, 6, 12, 24, 36, 80, 100, 150, 240, 560, 420, 660, 1020, 1680, 4032, 1764, 2940, 4620, 7224, 12096, 29568, 7392, 13104, 21280, 33320, 52416, 88704, 219648, 30888, 58212, 98280, 156870, 244800, 386496, 658944, 1647360, 128700, 257400, 452760, 742140, 1170540, 1821600, 2882880, 4942080, 12446720

Offset: 1

Michel Marcus, Mar 01 2019

			Triangle begins
     2,
     6,   12,
    24,   36,   80,
   100,  150,  240,  560,
   420,  660, 1020, 1680,  4032,
  1764, 2940, 4620, 7224, 12096, 29568,
  ...

R. T. Eakin, A combinatorial partition of Mersenne numbers arising from spectroscopy, Journal of Number Theory, Volume 132, Issue 10, October 2012, Pages 2166-2183.

Sum of n-th row equals A000984(n)*A000225(n).

Right diagonal is A069723 starting at index 2.

T(n,r) = binomial(2*n-2*r,n-r)*((n+1)/r)*sum(k=0, (r-1)\2, (-1)^k*binomial(2*r,k)*binomial(n+3*r-2*k,r-2*k-1));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")););

cp's OEIS Frontend

A258758 Triangle T(n,k) = C(n+k-1,k)*C(2*n-1,n-k).

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Magma

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A372506 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-2*x) )^n.

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Maple

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A128406 a(n) = (n+1)*2^(n*(n+1)).

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A306625 Regular triangle T(n,k) = binomial(2*n-2*k,n-k)*((n+1)/k)*Sum_{k=0..floor((k-1)/2)} (-1)^k*binomial(2*k,k)*binomial(n+3*k-2*k,k-2*k-1), read by rows.

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PARI

A258758 Triangle T(n,k) = C(n+k-1,k)C(2n-1,n-k).

A128406 a(n) = (n+1)2^(n(n+1)).

A306625 Regular triangle T(n,k) = binomial(2n-2k,n-k)((n+1)/k)Sum_{k=0..floor((k-1)/2)} (-1)^kbinomial(2k,k)binomial(n+3k-2k,k-2k-1), read by rows.