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Search: id:A089900
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| A089900 |
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Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the factorials, starting with row 0: {1!,2!,3!,...}. |
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+0
4
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| 1, 2, 1, 6, 3, 1, 24, 11, 4, 1, 120, 49, 18, 5, 1, 720, 261, 92, 27, 6, 1, 5040, 1631, 536, 159, 38, 7, 1, 40320, 11743, 3552, 1029, 256, 51, 8, 1, 362880, 95901, 26608, 7353, 1848, 389, 66, 9, 1, 3628800, 876809, 223456, 58095, 14384, 3125, 564, 83, 10, 1
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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Row 1 is A001339, antidiagonal sums form A089902 and the main diagonal is A089901; the next lower diagonal forms {1,4,27,256,..,n^n,..}, which is the hyperbinomial transform (cf. A088956) of the main diagonal.
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FORMULA
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T(0, k)=(k+1)!, T(n+1, n)=(n+1)^(n+1), T(n, k)=sum_{i=0..k}n^(k-i)*binomial(k, i)*(i+1)!
E.g.f.: 1/((1-y*exp(x))*(1-x)^2). E.g.f. (n-th row): exp(n*x)/(1-x)^2.
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EXAMPLE
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Note secondary diagonal: {(n+1)^(n+1)}; rows begin:
1, 2,. 6,. 24,. 120,.. 720,.. 5040,..
1, 3, 11,. 49,. 261,. 1631,. 11743,..
1,_4, 18,. 92,. 536,. 3552,. 26608,..
1, 5,_27, 159, 1029,. 7353,. 58095,..
1, 6, 38,_256, 1848, 14384, 121264,..
1, 7, 51, 389,_3125, 26595, 241015,..
1, 8, 66, 564, 5016,_46656, 456048,..
1, 9, 83, 787, 7701, 78077,_823543,..
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PROGRAM
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(PARI) T(n, k)=if(n<0|k<0, 0, sum(i=0, k, n^(k-i)*binomial(k, i)*(i+1)!))
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CROSSREFS
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Cf. A001339, A088956, A089901, A089902.
Sequence in context: A163269 A103905 A103209 this_sequence A138533 A096334 A107867
Adjacent sequences: A089897 A089898 A089899 this_sequence A089901 A089902 A089903
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Nov 14 2003
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