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A288245
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Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=7 data values.
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7
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1, 7, -6, 49, -63, 15, 343, -588, 140, 126, -20, 2401, -5145, 1225, 2205, -175, -525, 15, 16807, -43218, 10290, 27783, -1470, -8820, 126, -2646, 630, 525, -6, 117649, -352947, 84035, 302526, -12005, -108045, 1029, -64827, 10290, 8575, -49, 15435, -441, -1225, 1
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OFFSET
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1,2
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COMMENTS
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Let SM_k = Sum( d_(t_1, t_2, t_3, ..., t_7)* eM_1^t_1 * eM_2^t_2 * ... * eM_7^t_7) summed over all length 7 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + ... + 7*t_7 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 7 data (i.e., SM_k = S_k/7 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(7,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_7) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.
Row sums of positive entries give 1,7,64,609,5846,56161,... Row sums of negative entries are always 1 less than corresponding row sums of positive entries.
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LINKS
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EXAMPLE
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Triangular array begins...
1;
7,-6;
49,-63,15;
343,-588,140,126,-20;
2401,-5145,1225,2205,-175,-525,15;
16807,-43218,10290,27783,-1470,-8820,126,-2646,630,525,-6;
117649,-352947,84035,302526,-12005,-108045,1029,64827,10290,8575,-49,15435,-441,-1225,1;
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PROG
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(Java) see links
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CROSSREFS
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First entries of each row of triangle are powers of m=7, A000420.
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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