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 A202476 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + k*x + x^2). 4
 1, 1, 2, 5, 11, 28, 74, 206, 601, 1826, 5766, 18851, 63676, 221678, 793958, 2920292, 11014653, 42543773, 168074091, 678403932, 2794920078, 11742254750, 50266213000, 219085792538, 971543475593, 4380664101448, 20071848941411, 93403455862117, 441206005123701 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..300 FORMULA Antidiagonal sums of the irregular triangle in which row n is defined by the g.f.: Product_{k=1..n} (1 + k*x + x^2) for n>=0. G.f.: 1/(1 - x*(1+x+x^2)/(1 + x*(1+x+x^2) - x*(1+2*x+x^2)/(1 + x*(1+2*x+x^2) - x*(1+3*x+x^2)/(1 + x*(1+3*x+x^2) - x*(1+4*x+x^2)/(1 + x*(1+4*x+x^2) -...))))), a continued fraction. EXAMPLE The coefficients in Product_{k=1..n} (1+k*x+x^2), n>=0, form the triangle: [1]; [1, 1, 1]; [1, 3, 4, 3, 1]; [1, 6, 14, 18, 14, 6, 1]; [1, 10, 39, 80, 100, 80, 39, 10, 1]; [1, 15, 90, 285, 539, 660, 539, 285, 90, 15, 1]; [1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 840, 181, 21, 1]; [1, 28, 329, 2128, 8400, 21392, 36630, 43624, 36630, 21392, 8400, 2128, 329, 28, 1]; ... the antidiagonal sums of which form this sequence. PROG (PARI) {a(n)=sum(k=0, n, polcoeff(prod(j=1, n-k, 1+j*x+x^2), k))} (PARI) {a(n)=local(CF=1+x+x*O(x^n)); for(k=1, n-1, CF=(1+(n-k)*x+x^2)/(1 + x*(1+(n-k)*x+x^2) - x*CF+x*O(x^n))); polcoeff(1/(1-x*CF), n)} CROSSREFS Cf. A201951, A201826, A202474. Sequence in context: A121398 A359181 A000625 * A210517 A244531 A288390 Adjacent sequences: A202473 A202474 A202475 * A202477 A202478 A202479 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 19 2011 STATUS approved

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Last modified September 22 10:21 EDT 2023. Contains 365520 sequences. (Running on oeis4.)