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 A300785 Triangle read by rows: T(n,k) = 140*k^3*(n-k)^3 - 14*k*(n-k)+1; n >= 0, 0 <= k <= n. 9
 1, 1, 1, 1, 127, 1, 1, 1093, 1093, 1, 1, 3739, 8905, 3739, 1, 1, 8905, 30157, 30157, 8905, 1, 1, 17431, 71569, 101935, 71569, 17431, 1, 1, 30157, 139861, 241753, 241753, 139861, 30157, 1, 1, 47923, 241753, 472291, 573217, 472291, 241753, 47923, 1, 1, 71569, 383965, 816229, 1119721, 1119721, 816229, 383965, 71569, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The triangle is symmetric: T(n,k) = T(n,n-k). From Kolosov Petro, Oct 22 2018: (Start) Let R(n,k) := A302971(n,k)/A304042(n,k). T(n,k) is the case m = 1 in L(m,n,k) := Sum_{j=0..m} R(n,k) * k^j * (n-k)^j. For the cases m = 1 and m = 2 see A287326 and A300656, respectively. For l = n, S1(m,l,n) = Sum_{k=1..l} Sum_{j=0..m} R(n,k) * k^j * (n-k)^j. The result is: S(m,n,n) = n^(2m+1) for every positive integer m and n. Also, for l = n, S0(m,l-1,n) = Sum_{k=0..l-1} Sum_{j=0..m} R(n,k) * k^j * (n-k)^j. The result is: S(m,n,n) = n^(2m+1) for every positive integer m and n. (End) The term k^j * (n-k)^j in above formulae is A094053(n,k)^j. - Kolosov Petro, Dec 13 2018 LINKS Muniru A Asiru, Rows n=0..100 of triangle, flattened. Petro Kolosov, Another Power Identity involving Binomial Theorem and Faulhaber's formula, arXiv:1603.02468 [math.NT], 2016-2018. FORMULA T(n,k) = 140*k^3*(n-k)^3 - 14*k*(n-k) + 1. T(n+3,k) = 4*T(n+2,k) - 6*T(n+1,k) + 4*T(n,k) - T(n-1,k), for n >= k. Sum_{k=1..n} T(n,k) = A001015(n). Sum_{k=0..n-1} T(n,k) = A001015(n). From Kolosov Petro, Dec 13 2018: (Start) Sum_{k=1..r} T(n, k) = -A316387(3,r,0)*n^0 + A316387(3,r,1)*n^1 - A316387(3,r,2)*n^2 + A316387(3,r,3)*n^3. Sum_{k=0..n} T(n,k) = A258806(n). Sum_{k=1..n-1} T(n,k) = A258808(n). Sum_{k=1..n-1} T(n,k) = -A024005(n). T(n,k) = 140*A094053(n,k)^3 + 0*A094053(n,k)^2 - 14*A094053(n,k)^1 + 1*A094053(n,k)^0.(End) EXAMPLE Triangle begins: -------------------------------------------------------------------------- k =     0      1       2       3        4        5       6       7      8 -------------------------------------------------------------------------- n = 0:  1; n = 1:  1,     1; n = 2:  1,   127,      1; n = 3:  1,  1093,   1093,      1; n = 4:  1,  3739,   8905,   3739,       1; n = 5:  1,  8905,  30157,  30157,    8905,       1; n = 6:  1, 17431,  71569, 101935,   71569,   17431,      1; n = 7:  1, 30157, 139861, 241753,  241753,  139861,  30157,      1; n = 8:  1, 47923, 241753, 472291,  573217,  472291, 241753,  47923,     1; MAPLE T:=(n, k)->140*k^3*(n-k)^3-14*k*(n-k)+1: seq(seq(T(n, k), k=0..n), n=0..9); # Muniru A Asiru, Dec 14 2018 MATHEMATICA (* Define the R[n, k] := (A302971/A304042)(n, k) *) R[n_, k_] := 0 R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*    Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*    BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n; (* Define the zero to power zero to be 1 *) Unprotect[Power]; Power[0 | 0., 0 | 0.] = 1; Protect[Power]; (* Define the sum Sum_{j=0..m} R(m, j) * k^j * (T-k)^j *) P[m_, T_, k_] := Sum[R[m, j]*k^j*(T - k)^j, {j, 0, m}]; (* Define the value of 'm' to be m = 3 for A300785 *) m=3; (* Print ten initial rows of triangle T(n, k) *) Column[Table[P[m, T, k], {T, 0, 10}, {k, 0, T}], Center] (* Kolosov Petro, Oct 06 2018 *) PROG (PARI) t(n, k) = 140*k^3*(n-k)^3-14*k*(n-k)+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) (MAGMA) /* As triangle */ [[140*k^3*(n-k)^3-14*k*(n-k)+1: k in [0..n]]: n in [0..10]]; // Bruno Berselli, Mar 21 2018 (Sage) [[140*k^3*(n-k)^3 - 14*k*(n-k)+1 for k in range(n+1)] for n in range(12)] # G. C. Greubel, Dec 14 2018 (GAP) T:=Flat(List([0..9], n->List([0..n], k->140*k^3*(n-k)^3 - 14*k*(n-k)+1))); # G. C. Greubel, Dec 14 2018 CROSSREFS Sequences A287326, A300656 represent the cases for m = 1, 2 in L(m,n,k), see comments, line 2. Row sums give A258806. Cf. A000584, A287326, A007318, A077028, A294317, A068236, A300656, A302971, A304042, A001015, A094053, A258808, A024005, A316387. Sequence in context: A025037 A281478 A212927 * A051335 A186995 A145586 Adjacent sequences:  A300782 A300783 A300784 * A300786 A300787 A300788 KEYWORD nonn,tabl,easy AUTHOR Kolosov Petro, Mar 12 2018 STATUS approved

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Last modified December 5 10:45 EST 2019. Contains 329751 sequences. (Running on oeis4.)