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A077042
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Square array read by antidiagonals of central polynomial coefficients: largest coefficient in expansion of (1+x+x^2+...+x^(n-1))^k=((1-x^n)/(1-x))^k, i.e. the coefficient of x^floor[k*(n-1)/2] and of x^ceiling[k*(n-1)/2]; also number of compositions of [k*(n+1)/2] into exactly k positive integers each no more than n.
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8
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1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 1, 1, 0, 1, 6, 7, 4, 1, 1, 0, 1, 10, 19, 12, 5, 1, 1, 0, 1, 20, 51, 44, 19, 6, 1, 1, 0, 1, 35, 141, 155, 85, 27, 7, 1, 1, 0, 1, 70, 393, 580, 381, 146, 37, 8, 1, 1, 0, 1, 126, 1107, 2128, 1751, 780, 231, 48, 9, 1, 1, 0, 1, 252, 3139
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,13
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LINKS
| Index entries for sequences related to compositions
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FORMULA
| By the central limit theorem, T(n, k) is roughly n^(k-1)*sqrt(6/(pi*k)).
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EXAMPLE
| Rows start:
1,0,0,0,0,0,0,...;
1,1,1,1,1,1,1,...;
1,1,2,3,6,10,20,...;
1,1,3,7,19,51,141,...;
1,1,4,12,44,155,580,...;
1,1,5,19,85,381,1751,...;
etc.
1;
0,1;
0,1,1;
0,1,1,1;
0,1,2,1,1;
0,1,3,3,1,1;
0,1,6,7,4,1,1;
...
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MATHEMATICA
| t[n_, k_] := Max[ CoefficientList[ Series[ ((1-x^n) / (1-x))^k, {x, 0, k*(n-1)}], x]]; t[0, 0] = 1; t[0, _] = 0; Flatten[ Table[ t[n-k, k], {n, 0, 12}, {k, n, 0, -1}]] (* From Jean-François Alcover, Nov 04 2011 *)
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PROG
| (PARI) T(n, k)=if(n<1|k<1, k==0, vecmax(Vec(((1-x^n)/(1-x))^k)))
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CROSSREFS
| Rows include A000007, A000012, A001405, A002426, A005190, A005191, A018901, A025012, A025013, A025014, A025015. Columns include A000012, A000012, A001477, A077043, A005900, A077044, A071816. Central diagonal is A077045, with A077046 and A077047 either side. Cf. A067059.
Sequence in context: A029387 A070878 A060959 * A144903 A108934 A108947
Adjacent sequences: A077039 A077040 A077041 * A077043 A077044 A077045
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KEYWORD
| nonn,tabl
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Oct 22 2002
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