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Search: id:A094728
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| A094728 |
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Triangle read by rows: T(n,k) = n^2 - k^2, 0<=k<n. |
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5
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| 1, 4, 3, 9, 8, 5, 16, 15, 12, 7, 25, 24, 21, 16, 9, 36, 35, 32, 27, 20, 11, 49, 48, 45, 40, 33, 24, 13, 64, 63, 60, 55, 48, 39, 28, 15, 81, 80, 77, 72, 65, 56, 45, 32, 17, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19, 121, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21, 144
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OFFSET
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1,2
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COMMENT
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T(n,0)=A000290(n); T(n,1)=A005563(n-1) for n>1; T(n,2)=A028347(n) for n>2; T(n,3)=A028560(n-3) for n>3; T(n,4)=A028566(n-4) for n>4;
T(n,n-1)=A005408(n); T(n,n-2)=A008586(n-1) for n>1; T(n,n-3)=A016945(n-2) for n>2; T(n,n-4)=A008590(n-2) for n>3; T(n,n-5)=A017329(n-3) for n>4; T(n,n-6)=A008594(n-3) for n>5; T(n,n-8)=A008598(n-2) for n>7;
T(A005408(k),k) = A000567(k);
row sums give A002412;
(T(n,k) mod 4) <> 2, see A042965, A016825;
all numbers m occur A034178(m) times;
T(n,k) = A004736(n,k)*A094727(n,k).
The row polynomials T(n,x) appear in the calculation of the column g.f.s of triangle A120070 (used to find the frequencies of the spectral lines of the hydrogen atom).
A093995 (1, 4, 4) - A143844 (0, 0, 1) ; a(n)= A143813 (or 1, A120070) especially mixed with (from 2) n^2. [From Paul Curtz (bpcrtz(AT)free.fr), Sep 06 2008]
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FORMULA
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Row polynomials: T(n,x) = n^2*sum(x^m,m=0..n)-sum(m^2*x^m,m=0..n) = sum(T(n,k)*x^k,k=0..n-1), n>=1.
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EXAMPLE
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n=3: T(3,x) = 9+8*x+5*x^2
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CROSSREFS
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Sequence in context: A010655 A123596 A094885 this_sequence A131805 A103218 A107381
Adjacent sequences: A094725 A094726 A094727 this_sequence A094729 A094730 A094731
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KEYWORD
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nonn,tabl
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AUTHOR
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Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 24 2004
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