A172021 Start with the triangle A171661, reverse its rows, add missing powers of 2 at beginning of each row.
1, 1, 2, 2, 1, 2, 4, 6, 6, 1, 2, 4, 8, 14, 20, 20, 1, 2, 4, 8, 16, 30, 50, 70, 70, 1, 2, 4, 8, 16, 32, 62, 112, 182, 252, 252, 1, 2, 4, 8, 16, 32, 64, 126, 238, 420, 672, 924, 924, 1, 2, 4, 8, 16, 32, 64, 128, 254, 492, 912, 1584, 2508, 3432, 3432
Offset: 1
Examples
Triangle begins: ......1 ....1,2,2 ..1,2,4,6,6 1,2,4,8,14,20,20 From _Mark Dols_, Jan 24 2010: (Start) Interspaced binomial transform of 1^n: 1...1...1...1...1...1... ..2...2...2...2...2...2. 2...4...4...4...4...4... ..6...8...8...8...8...8. 6.. 14..16..16..16..16.. ..20..30..32..32..32..32 20..50..62..64..64..64.. (End)
Programs
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Derive
T(n,k):=POLY_COEFF(SUM(x^i/i!, i, 0, n)^2, x, k)·k! TABLE(VECTOR(T(v, u), u, 0, 2·v), v, 0, 10) # Giovanni Artico, Aug 30 2013
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Maple
seq(PolynomialTools:-CoefficientList((convert(taylor(exp(x),x,n+1),polynom)^2),x)*~[seq(i!,i=0..2 n)],n=0..10) # Giovanni Artico, Aug 30 2013
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Mathematica
Table[CoefficientList[Series[(Sum[x^i/i!, {i, 0, m}])^2, {x, 0, 2 m}], x]*Table[n!, {n, 0, 2 m}], {m, 0, 10}] // Grid (* Geoffrey Critzer, Mar 15 2010 *)
Formula
E.g.f. for row n is: ( 1 + x + x^2/2! + ... + x^n/n! )^2. - Geoffrey Critzer, Mar 15 2010
Extensions
Definition rewritten by N. J. A. Sloane, Jan 23 2010
More terms from Mark Dols, Jan 24 2010
Comments