A034430 -id:A034430 - cp's OEIS frontend

A108032 Triangle T(n,k), 0<=k<=n, read by rows, defined by : T(0,0) = 1, T(n,k) = 0 if n

1, 1, 1, 3, 2, 2, 15, 9, 6, 6, 105, 60, 36, 24, 24, 945, 525, 300, 180, 120, 120, 10395, 5670, 3150, 1800, 1080, 720, 720, 135135, 72765, 39690, 22050, 12600, 7560, 5040, 5040, 2027025, 1081080, 582120, 317520, 176400, 100800, 60480, 40320, 40320

Offset: 0

Philippe Deléham, Jun 01 2005

			1;
1, 1;
3, 2, 2;
15, 9, 6, 6;
105, 60, 36, 24, 24; ...

Diagonals : A001147, A001193, A000142.

Cf. A034430 (row sums).

Sum{ k, 0<=k<=n} T(n, k) = A034430(n).
T(n, k) = A001147(n-k)*k!*binomial(n, k).
E.g.f.: 1/(1-t*x)*1/sqrt(1-2*x) = 1 + x*(1+t) + x^2/2!*(3+2*t+2*t^2) + .... - Peter Bala, Jun 27 2012

A331403 E.g.f.: 1 / ((1 + x) * sqrt(1 - 2*x)).

Original entry on oeis.org

1, 0, 3, 6, 81, 540, 7155, 85050, 1346625, 22339800, 431331075, 9004668750, 208178118225, 5199538043700, 140664514065075, 4080315642653250, 126613733680058625, 4180226398201854000, 146399020309066399875, 5419213146765629961750, 211446723837565171580625

Offset: 0

Ilya Gutkovskiy, Jan 16 2020

nonn

Cf. A001147, A005359, A034430, A052585, A317618.

nmax = 20; CoefficientList[Series[1/((1 + x) Sqrt[1 - 2 x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^(n - k) (2 k - 1)!!/k!, {k, 0, n}], {n, 0, 20}]

a(n) = {n! * sum(k=0, n, (-1)^(n - k) * (2*k)! / (2^k*k!^2))} \\ Andrew Howroyd, Jan 16 2020

seq(n) = {Vec(serlaplace(1 / ((1 + x) * sqrt(1 - 2*x + O(x*x^n)))))} \\ Andrew Howroyd, Jan 16 2020

a(n) = n! * Sum_{k=0..n} (-1)^(n - k) * (2*k - 1)!! / k!.
D-finite with recurrence: a(n) +(-n+1)*a(n-1) -(2*n-1)*(n-1)*a(n-2)=0. - R. J. Mathar, Jan 25 2020
a(n) ~ 2^(n + 3/2) * n^n / (3*exp(n)). - Vaclav Kotesovec, Jan 26 2020

cp's OEIS Frontend

A108032 Triangle T(n,k), 0<=k<=n, read by rows, defined by : T(0,0) = 1, T(n,k) = 0 if n

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