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A084868
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Main diagonal of symmetric square table A084867, in which the antidiagonal sums (A006012) form the first row shifted left.
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3
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1, 2, 8, 36, 168, 796, 3800, 18216, 87536, 421292, 2029592, 9784088, 47187536, 227651352, 1098523504, 5301727824, 25590307552, 123529362124, 596337248024, 2878947861432, 13899229883024, 67105641925064, 323993230750672
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The Hankel transform (see A001906 for definition) of this sequence is A000302 (powers of 4) : 1, 4, 16, 64, 256, 1024, ... - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 17 2005
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FORMULA
| Differential equation: (16*x^3 + 12*x^2 - 8*x + 1) * x*diff(A(x), x) + (8x^3 - 12*x^2 + 6*x - 1) * A(x) + (8x^2 - 6*x + 1) = 0.
G.f.: ((1 - 4*x) + 2*x * sqrt(1 - 4*x)) / (1 - 4*x - 4*x^2). a(n) * (n-1) = a(n-1) * (8*n - 14) - a(n-2) * 12*(n-3) - a(n-3) * 8*(2*n - 5), n>2. Hankel number wall zig-zag diagonal is A011782. - Michael Somos, Sep 14 2003
INVERT transform of A028329 (offset 1). - Michael Somos, Jan 05 2012
G.f.: (1-2*x*f(x))/(1-2*x*f(x)-2*x) where f(x) is the g.f. of A000108 (Catalan numbers). - DELEHAM Philippe, Jan 30 2012
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EXAMPLE
| 1 + 2*x + 8*x^2 + 36*x^3 + 168*x^4 + 796*x^5 + 3800*x^6 + 18216*x^7 + ...
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PROG
| (PARI) {a(n) = if( n<0, 0, polcoeff((1 - 4*x + 2*x * sqrt(1 - 4*x + x * O(x^n))) /(1 - 4*x - 4*x^2), n))} /* Michael Somos, Jan 05 2012 */
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CROSSREFS
| Cf. A006012, A011782, A028329, A084867.
Sequence in context: A147722 A089387 A206902 * A109980 A186338 A190862
Adjacent sequences: A084865 A084866 A084867 * A084869 A084870 A084871
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jun 10 2003, Jun 11 2003
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