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A084785 Diagonal of the triangle (A084783) and the self-convolution of the first column (A084784). 7
1, 2, 5, 16, 66, 348, 2298, 18504, 176841, 1958746, 24661493, 347548376, 5415830272, 92410046544, 1712819553864, 34258146124320, 735267392077962, 16852848083339700, 410809882438699346, 10611174406149372736 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

In the triangle (A084783), the diagonal (this sequence) is the self-convolution of the first column (A084784) and the row sums (A084786) gives the differences of the diagonal and the first column.

REFERENCES

Chao-Ping Chen, Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant, Journal of Number Theory, 2016, Volume 172, March 2017, Pages 145-159; https://doi.org/10.1016/j.jnt.2016.08.010

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

FORMULA

G.f. A(x) satisfies (1+x)^2 = A(x/(1+x))^2/A(x). - Michael Somos, Feb 16 2006

G.f.: A(x) = Product_{n>=1} 1/(1 - n*x)^(1/2^n). - Paul D. Hanna, Jun 16 2010

a(n) ~ (n-1)! / (log(2))^(n+1). - Vaclav Kotesovec, Nov 19 2014

From Peter Bala, May 26 2001: (Start)

O.g.f. A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ), where b(n) = (-1)^n*Sum_{k = 1..n} k!*Stirling2(n,k)*(-2)^k = A000629(n) = 2*A000670(n) for n >= 1. Cf. A090352.

sqrt(A(x)) = 1/(1 + x)*A(x/(1 + x)) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 137*x^5 + ... is the o.g.f. for A084784. See also A019538. (End)

EXAMPLE

G.f.: A(x) = (1-x)^(-1/2)*(1-2*x)^(-1/4)*(1-3*x)^(-1/8)*(1-4*x)^(-1/16)*... - Paul D. Hanna, Jun 16 2010

PROG

(PARI) A = matrix(25, 25); A[1, 1] = 1; rs = 1; print(1); for (n = 2, 25, sc = sum (i = 2, n - 1, A[i, 1]*A[n + 1 - i, 1]); A[n, 1] = rs - sc; rs = A[n, 1]; for (k = 2, n, A[n, k] = A[n, k - 1] + A[n - 1, k - 1]; rs += A[n, k]); print(A[n, n])); (Wasserman)

(PARI) {a(n)=local(A); if(n<0, 0, A=1; for(k=1, n, A=truncate(A+O(x^k))+x*O(x^k); A+=A-(subst(1/A, x, x/(1+x))*(1+x))^-2; ); polcoeff(A, n))} /* Michael Somos, Feb 18 2006 */

CROSSREFS

Cf. A084783, A084784, A084786; A019538, A090352.

Sequence in context: A182290 A007469 A091139 * A124551 A005157 A019502

Adjacent sequences:  A084782 A084783 A084784 * A084786 A084787 A084788

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Jun 13 2003

EXTENSIONS

More terms from David Wasserman, Jan 06 2005

STATUS

approved

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Last modified February 20 14:37 EST 2018. Contains 299380 sequences. (Running on oeis4.)