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 A300656 Triangle read by rows: T(n,k) = 30*k^2*(n-k)^2 + 1; n >= 0, 0 <= k <= n. 10
 1, 1, 1, 1, 31, 1, 1, 121, 121, 1, 1, 271, 481, 271, 1, 1, 481, 1081, 1081, 481, 1, 1, 751, 1921, 2431, 1921, 751, 1, 1, 1081, 3001, 4321, 4321, 3001, 1081, 1, 1, 1471, 4321, 6751, 7681, 6751, 4321, 1471, 1, 1, 1921, 5881, 9721, 12001, 12001, 9721, 5881, 1921, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS From Kolosov Petro, Apr 12 2020: (Start) Let be A(m, r) = A302971(m, r) / A304042(m, r). Let be L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r. Then T(n, k) = L(2, n, k). Fifth power can be expressed as row sum of triangle T(n, k). T(n, k) is symmetric: T(n, k) = T(n, n-k). (End) LINKS Kolosov Petro, Rows n = 0..2078 of triangle, flattened. Kolosov Petro, On the link between Binomial Theorem and Discrete Convolution of Power Function, arXiv:1603.02468 [math.NT], 2016-2020. FORMULA From Kolosov Petro, Apr 12 2020: (Start) T(n, k) = 30 * k^2 * (n-k)^2 + 1. T(n, k) = 30 * A094053(n,k)^2 + 1. T(n, k) = A158558((n-k) * k). T(n+2, k) = 3*T(n+1, k) - 3*T(n, k) + T(n-1, k), for n >= k. Sum_{k=1..n} T(n, k) = A000584(n). Sum_{k=0..n-1} T(n, k) = A000584(n). Sum_{k=0..n} T(n, k) = A002561(n). Sum_{k=1..n-1} T(n, k) = A258807(n). Sum_{k=1..n-1} T(n, k) = -A024003(n), n > 1. Sum_{k=1..r} T(n, k) = A316349(2,r,0)*n^0 - A316349(2,r,1)*n^1 + A316349(2,r,2)*n^2. (End) G.f.: (1 + 26*y + 336*y^2 + 326*y^3 + 31*y^4 + x^2*(1 + 116*y + 486*y^2 + 116*y^3 + y^4) + x*(-2 - 82*y - 882*y^2 - 502*y^3 + 28*y^4))/((-1 + x)^3*(-1 + y)^5). - Stefano Spezia, Oct 30 2018 EXAMPLE Triangle begins: -------------------------------------------------------------------------- k=    0     1     2      3      4      5      6      7     8     9    10 -------------------------------------------------------------------------- n=0:  1; n=1:  1,    1; n=2:  1,   31,    1; n=3:  1,  121,  121,     1; n=4:  1,  271,  481,   271,     1; n=5:  1,  481, 1081,  1081,   481,     1; n=6:  1,  751, 1921,  2431,  1921,   751,     1; n=7:  1, 1081, 3001,  4321,  4321,  3001,  1081,     1; n=8:  1, 1471, 4321,  6751,  7681,  6751,  4321,  1471,    1; n=8:  1, 1921, 5881,  9721, 12001, 12001,  9721,  5881, 1921,    1; n=10: 1, 2431, 7681, 13231, 17281, 18751, 17281, 13231, 7681, 2431,   1; MAPLE a:=(n, k)->30*k^2*(n-k)^2+1: seq(seq(a(n, k), k=0..n), n=0..9); # Muniru A Asiru, Oct 24 2018 MATHEMATICA T[n_, k_] := 30 k^2 (n - k)^2 + 1; Column[ Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Apr 12 2020 *) f[n_]:=Table[SeriesCoefficient[(1 + 26 y + 336 y^2 + 326 y^3 + 31 y^4 + x^2 (1 + 116 y + 486 y^2 + 116 y^3 + y^4) + x (-2 - 82 y - 882 y^2 - 502 y^3 + 28 y^4))/((-1 + x)^3 (-1 + y)^5), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Flatten[Array[f, 11, 0]] (* Stefano Spezia, Oct 30 2018 *) PROG (PARI) t(n, k) = 30*k^2*(n-k)^2+1 trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print("")) /* Print initial 9 rows of triangle as follows */ trianglerows(9) (GAP) T:=Flat(List([0..9], n->List([0..n], k->30*k^2*(n-k)^2+1))); # Muniru A Asiru, Oct 24 2018 (MAGMA) [[30*k^2*(n-k)^2+1: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Dec 14 2018 (Sage) [[30*k^2*(n-k)^2+1 for k in range(n+1)] for n in range(12)] # G. C. Greubel, Dec 14 2018 CROSSREFS Various cases of L(m, n, k): A287326(m=1), This sequence (m=2), A300785(m=3). See comments for L(m, n, k). Row sums give the nonzero terms of A002561. Cf. A000584, A287326, A007318, A077028, A294317, A068236, A302971, A304042, A002561, A258807, A158558, A094053, A024003, A316349. Sequence in context: A040963 A040962 A040961 * A239633 A174692 A172302 Adjacent sequences:  A300653 A300654 A300655 * A300657 A300658 A300659 KEYWORD nonn,tabl,easy AUTHOR Kolosov Petro, Mar 10 2018 STATUS approved

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Last modified September 24 17:09 EDT 2020. Contains 337321 sequences. (Running on oeis4.)