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 A132458 Let df(n,k) = Product_{i=0..k-1} (n-i) be the descending factorial and let P(m,n) = df(n-1,m-1)^2*(2*n-m)/((m-1)!*m!). Sequence gives P(4,n). 7
 0, 0, 0, 1, 24, 200, 1000, 3675, 10976, 28224, 64800, 136125, 266200, 490776, 861224, 1449175, 2352000, 3699200, 5659776, 8450649, 12346200, 17689000, 24901800, 34500851, 47110624, 63480000, 84500000, 111223125, 144884376, 186924024, 239012200, 303075375 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS P(m,n) is the number of n-step paths that start from (0,0) and reach (m,m) for the first time, where the steps are of the following 4 types: N=(x,y)->(x,y+1), E=(x,y)->(x+1,y), NE=(x,y)->(x+1,y+1), LOOP=(x,y)->(x,y). For m = 1 through 8 we get respectively A005408, A000578, A108674, this sequence, A133317, A132464, A132465, A132466. LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1). FORMULA From Bruno Berselli, Dec 29 2010: (Start) a(n) = (n-3)^2*(n-2)^3*(n-1)^2/72. G.f.: x^4*(1+16*x+36*x^2+16*x^3+x^4)/(1-x)^8. (End) Sum_{n>=4} 1/a(n) = 72*zeta(3) - 171/2. - Jaume Oliver Lafont, Aug 06 2017 MAPLE df:=proc(n, k) mul(n-i, i=0..k-1); end; P:=proc(n, k) df(k-1, n-1)^2*(2*k-n)/((n-1)!*n!); end; [seq(P(4, n), n=1..50)]; MATHEMATICA CoefficientList[Series[x^3 (1 + 16 x + 36 x^2 + 16 x^3 + x^4) / (1 - x)^8, {x, 0, 33}], x] (* Vincenzo Librandi, Aug 06 2017 *) PROG (MAGMA) [(n-3)^2*(n-2)^3*(n-1)^2/72: n in [1..40]]; // Vincenzo Librandi, Aug 06 2017 CROSSREFS Cf. A005408, A000578, A108674, A133317, A132464, A132465, A132466. Sequence in context: A198396 A225296 A048355 * A055857 A239574 A223748 Adjacent sequences:  A132455 A132456 A132457 * A132459 A132460 A132461 KEYWORD nonn AUTHOR Ottavio D'Antona (dantona(AT)dico.unimi.it), Oct 31 2007 STATUS approved

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Last modified October 17 11:26 EDT 2019. Contains 328108 sequences. (Running on oeis4.)