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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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4
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
| 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
| n=3: T(3,x) = 9+8*x+5*x^2
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CROSSREFS
| Sequence in context: A200361 A180858 A094885 * A131805 A197694 A103218
Adjacent sequences: A094725 A094726 A094727 * A094729 A094730 A094731
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KEYWORD
| nonn,tabl
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AUTHOR
| Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 24 2004
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