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 A129261 a(n) = coefficient of x^n in Product_{k=0..n} (1 + 2x + 3x^2 +...+ (k+1)*x^k). 1
 1, 2, 7, 36, 204, 1222, 7513, 46950, 296691, 1890232, 12118424, 78080402, 505134625, 3279051382, 21347213562, 139319046744, 911204289149, 5970941722698, 39192011365250, 257632856738690, 1695850232984011 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA a(n) = [x^n] Product_{k=1..n+1} { (1 - (k+1)*x^k + k*x^(k+1))/(1-x)^2 } for n>0, with a(0)=1. EXAMPLE a(2) = [x^2] (1 + 2x)*(1 + 2x + 3x^2) = [x^2] (1 + 4x + 7x^2 + 6x^3) = 7. a(3) = [x^3] (1 + 2x)*(1 + 2x + 3x^2)*(1 + 2x + 3x^2 + 4x^3) = [x^3] (1 + 6x + 18x^2 + 36x^3 + 49x^4 + 46x^5 + 24x^6) = 36. This sequence is a diagonal in the triangle of successive products: (1); 1,(2); 1,4,(7),6; 1,6,18,(36),49,46,24; 1,8,33,94,(204),354,497,562,501,326,120; 1,10,52,188,528,(1222),2406,4102,6116,7996,9132,9014,7541,5116,2556,720; ... Lower diagonals are convolutions with this sequence and A006013: [1,4,18,94,528,3106,18798,115964, ...] = A006013 * A129261; [1,6,33,188,1105,6660,40888,254510,...]= A006013^2 * A129261. PROG (PARI) {a(n)=polcoeff(prod(k=1, n+1, (1 - (k+1)*x^k + k*x^(k+1))/(1-x)^2), n)} CROSSREFS Cf. A006013. Sequence in context: A111908 A060814 A192816 * A177258 A018997 A018954 Adjacent sequences:  A129258 A129259 A129260 * A129262 A129263 A129264 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 06 2007 STATUS approved

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Last modified May 24 07:09 EDT 2013. Contains 225617 sequences.