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 A132945 Generalization of an a(n)=3*2^n*a(n-1) as 3=(m+1) and 2=m To give general term: t(n,m)=a(n)=(m+1)^n*m^(n*(n-1)/2). 0
 1, 1, 2, 1, 4, 18, 1, 8, 216, 1728, 1, 16, 5184, 186624, 2560000, 1, 32, 248832, 60466176, 3276800000, 75937500000, 1, 64, 23887872, 58773123072, 16777216000000, 1423828125000000, 55316793250381824, 1, 128, 4586471424 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS From the ratio: a[2*n+1]/a[n]=(p/q)^(2*n)/(1/q)^(2*n+1)=q*p^(2*n) where p/q+1/q=1 or q=p+1 to give a[2*n+1]=(p+1)*p^(2*n)*a[2*n) Substitution of 2*n+1=m gives: a[m]=(p+1)*p^(m-1)*a[m] The general term is: a[n]=(p+1)^n*p(n*n-1)/2) Tha generalizes to the triangular sequence: t{n,m]=(m+1)^n*m^(n*(n-1)/2) There are a sequence of integer sequences. The row sums are: Table[Apply[Plus, Table[If[m == 0, 1, (m + 1)^n*m^(n*(n - 1)/2)], {m, 0, n}]], {n, 0, 10}]; {1, 3, 23, 1953, 2751825, 79275015041, 56757457388392833, 1189555425750705365861377, 840397436855599094944934404993537, 22654481007347397384838896692805197152547329, 26024856222487890805745042737267875522571222279183006721} LINKS Table of n, a(n) for n=1..31. FORMULA If m==0,t(n,0)=1 else t(n,m)=a(n)=(m+1)^n*m^(n*(n-1)/2) EXAMPLE {1}, {1, 2}, {1, 4, 18}, {1, 8, 216, 1728}, {1, 16, 5184, 186624, 2560000}, {1, 32, 248832, 60466176, 3276800000, 75937500000}, {1, 64, 23887872, 58773123072, 16777216000000, 1423828125000000, 55316793250381824} MATHEMATICA a = Table[Table[If[m == 0, 1, (m + 1)^n*m^(n*(n - 1)/2)], {m, 0, n}], {n, 0, 10}]; Flatten[a] CROSSREFS Sequence in context: A238454 A025228 A267377 * A192494 A013156 A012925 Adjacent sequences: A132942 A132943 A132944 * A132946 A132947 A132948 KEYWORD nonn,tabl,uned AUTHOR Roger L. Bagula, Nov 19 2007 STATUS approved

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Last modified May 30 07:51 EDT 2023. Contains 363050 sequences. (Running on oeis4.)