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A100227 Main diagonal of triangle A100226. 4
1, 1, 5, 13, 33, 81, 197, 477, 1153, 2785, 6725, 16237, 39201, 94641, 228485, 551613, 1331713, 3215041, 7761797, 18738637, 45239073, 109216785, 263672645, 636562077, 1536796801, 3710155681, 8957108165, 21624372013, 52205852193, 126036076401, 304278004997 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Specify that a triangle has T(n,0)=T(n,n)=(n+1)*(n+2)/2.  The interior terms T(r,c)=T(r-1,c)+T(r-1,c-1)+T(r-2,c-1).  The difference between the sum of the terms in row(n+1) and those in row(n) is a(n+2). - J. M. Bergot, Mar 15 2013

Starting with offset 1 the sequence is A001333: (1, 3, 7, 17, 41,...), convolved with (1, 2, 0, 2, 0, 2,...). - Gary W. Adamson, Aug 10 2016

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1)

FORMULA

a(n) = A002203(n) - 1.

a(n) = 2*a(n-1) + a(n-2) + 2 for n>1, with a(0)=1, a(1)=1.

G.f.: Sum_{n>=1} a(n)*x^n/n = log((1-x)/(1-2*x-x^2)).

G.f.: (1-2*x+3*x^2)/((x-1)(x^2+2*x-1)). - Paul D. Hanna, Feb 22 2005

a(n) = n*sum(k=1..n, sum(i=0..n-k, binomial(k, i)*binomial(n-i-1, k-1))/k), n>0, a(0)=1. - Vladimir Kruchinin, May 13 2011

a(n) = (-1+(1-sqrt(2))^n+(1+sqrt(2))^n). - Colin Barker, Mar 16 2016

E.g.f.: (2*cosh(sqrt(2)*x) - 1)*exp(x). - Ilya Gutkovskiy, Aug 22 2016

PROG

(PARI) a(n)=if(n==0, 1, n*polcoeff(log((1-x)/(1-2*x-x^2)+x*O(x^n)), n))

(PARI) a(n)=polcoeff((1-2*x+3*x^2)/(1-3*x+x^2+x^3)+x*O(x^n), n)

(Maxima)

a(n):=if n=0 then 1 else n*sum(sum(binomial(k, i)*binomial(n-i-1, k-1), i, 0, n-k)/k, k, 1, n); \\ Vladimir Kruchinin, May 13 2011

CROSSREFS

Cf. A100225, A100226, A001333.

Sequence in context: A272828 A001981 A141025 * A185454 A278764 A183774

Adjacent sequences:  A100224 A100225 A100226 * A100228 A100229 A100230

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Nov 29 2004

STATUS

approved

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Last modified October 22 04:29 EDT 2018. Contains 316431 sequences. (Running on oeis4.)