A110321 -id:A110321 - cp's OEIS frontend

A110322 Row sums of A110321, a number triangle related to the Jacobsthal numbers.

1, 2, 9, 52, 425, 4206, 50737, 708464, 11350257, 204171130, 4084757561, 89849981772, 2156575777369, 56068679418662, 1569955094823585, 47098171778191816, 1507149193966389857, 51242941744764975474

Offset: 0

Paul Barry, Jul 20 2005

Row sums of number triangle A110321.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001045, A110321.

CoefficientList[Series[E^x/(1-x-2*x^2), {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* Vaclav Kotesovec, Oct 18 2012 *)

E.g.f.: exp(x)/(1-x-2*x^2).
a(n) = Sum_{k=0..n} n!*J(n-k+1)/k! where J(n)=A001045(n).
a(n) = Sum_{k=0..n} binomial(n, k)*k!*J(k+1) where J(n)=A001045(n).
a(n) ~ n!*2^(n+1)*exp(1/2)/3. - Vaclav Kotesovec, Oct 18 2012
Conjecture: a(n) +(-n-1)*a(n-1) -(2*n-1)*(n-1)*a(n-2) +2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 11 2014
a(n) - n*a(n-1) - 2*n*(n-1)*a(n-2) - 1 = 0. - Martin Clever, Mar 22 2023
a(n) = (2*e^(1/2)*2^n*Gamma(n+1,1/2)+e^-1*(-1)^n*Gamma(n+1,-1))/3. - Martin Clever, Mar 25 2023

A110324 Inverse of a number triangle related to the Jacobsthal numbers.

Original entry on oeis.org

1, -1, 1, -4, -2, 1, 0, -12, -3, 1, 0, 0, -24, -4, 1, 0, 0, 0, -40, -5, 1, 0, 0, 0, 0, -60, -6, 1, 0, 0, 0, 0, 0, -84, -7, 1, 0, 0, 0, 0, 0, 0, -112, -8, 1, 0, 0, 0, 0, 0, 0, 0, -144, -9, 1, 0, 0, 0, 0, 0, 0, 0, 0, -180, -10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -220, -11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -264, -12, 1

Offset: 0

Paul Barry, Jul 20 2005

Row sums are A110325. Diagonal sums are A110326. Inverse of A110321. The result can be generalized as follows: The triangle whose columns have e.g.f. (x^k/k!)/(1-a*x-b*x^2) has inverse T(n,k)=if(n=k,1,if(n-k=1,-a*binomial(n,1),if(n-k=2,-2*b*binomial(n,2),0)))

			Rows begin
1;
-1,1;
-4,-2,1;
0,-12,-3,1;
0,0,-24,-4,1;
0,0,0,-40,-5,1;
0,0,0,0,-60,-6,1;

T(n, k)=if(n=k, 1, if(n-k=1, -binomial(n, 1), if(n-k=2, -4*binomial(n, 2), 0)))

A110323 Diagonal sums of a number triangle related to the Jacobsthal numbers.

Original entry on oeis.org

1, 1, 7, 32, 283, 2643, 32317, 443824, 7115461, 127222805, 2541399091, 55739417376, 1335729425647, 34666455314407, 969419191792633, 29046504524166368, 928541094344685577, 31540972026299055849

Offset: 0

Paul Barry, Jul 20 2005

Diagonal sums of number triangle A110321.

J[n_] := (2^n-(-1)^n)/3; a[n_] := Sum[(n-k)!*J[n-2*k+1]/k!, {k, 0, n/2}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Oct 08 2016 *)

a(n)=sum{k=0..floor(n/2), (n-k)!*J(n-2k+1)/k!}, J(n)=A001045(n).

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A110322 Row sums of A110321, a number triangle related to the Jacobsthal numbers.

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A110324 Inverse of a number triangle related to the Jacobsthal numbers.

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A110323 Diagonal sums of a number triangle related to the Jacobsthal numbers.

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