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%I #149 Feb 09 2024 12:37:35
%S 1,1,1,1,31,1,1,121,121,1,1,271,481,271,1,1,481,1081,1081,481,1,1,751,
%T 1921,2431,1921,751,1,1,1081,3001,4321,4321,3001,1081,1,1,1471,4321,
%U 6751,7681,6751,4321,1471,1,1,1921,5881,9721,12001,12001,9721,5881,1921,1
%N Triangle read by rows: T(n,k) = 30*k^2*(n-k)^2 + 1; n >= 0, 0 <= k <= n.
%C From _Kolosov Petro_, Apr 12 2020: (Start)
%C Let A(m, r) = A302971(m, r) / A304042(m, r).
%C Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.
%C Then T(n, k) = L(2, n, k).
%C Fifth power can be expressed as row sum of triangle T(n, k).
%C T(n, k) is symmetric: T(n, k) = T(n, n-k). (End)
%H Kolosov Petro, <a href="/A300656/b300656.txt">Rows n = 0..2078 of triangle, flattened</a>.
%H Kolosov Petro, <a href="https://arxiv.org/abs/1603.02468">On the link between Binomial Theorem and Discrete Convolution of Power Function</a>, arXiv:1603.02468 [math.NT], 2016-2020.
%H Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/PolynomialIdentityInvolvingBTandFaulhaber.pdf">Polynomial identity involving binomial theorem and Faulhaber's formula</a>, 2023.
%H Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/HistoryAndOverviewOfPolynomialP.pdf">History and overview of the polynomial P_b^m(x)</a>, 2024.
%F From _Kolosov Petro_, Apr 12 2020: (Start)
%F T(n, k) = 30 * k^2 * (n-k)^2 + 1.
%F T(n, k) = 30 * A094053(n,k)^2 + 1.
%F T(n, k) = A158558((n-k) * k).
%F T(n+2, k) = 3*T(n+1, k) - 3*T(n, k) + T(n-1, k), for n >= k.
%F Sum_{k=1..n} T(n, k) = A000584(n).
%F Sum_{k=0..n-1} T(n, k) = A000584(n).
%F Sum_{k=0..n} T(n, k) = A002561(n).
%F Sum_{k=1..n-1} T(n, k) = A258807(n).
%F Sum_{k=1..n-1} T(n, k) = -A024003(n), n > 1.
%F 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)
%F 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
%e Triangle begins:
%e --------------------------------------------------------------------------
%e k= 0 1 2 3 4 5 6 7 8 9 10
%e --------------------------------------------------------------------------
%e n=0: 1;
%e n=1: 1, 1;
%e n=2: 1, 31, 1;
%e n=3: 1, 121, 121, 1;
%e n=4: 1, 271, 481, 271, 1;
%e n=5: 1, 481, 1081, 1081, 481, 1;
%e n=6: 1, 751, 1921, 2431, 1921, 751, 1;
%e n=7: 1, 1081, 3001, 4321, 4321, 3001, 1081, 1;
%e n=8: 1, 1471, 4321, 6751, 7681, 6751, 4321, 1471, 1;
%e n=9: 1, 1921, 5881, 9721, 12001, 12001, 9721, 5881, 1921, 1;
%e n=10: 1, 2431, 7681, 13231, 17281, 18751, 17281, 13231, 7681, 2431, 1;
%p 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
%t T[n_, k_] := 30 k^2 (n - k)^2 + 1; Column[
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* _Kolosov Petro_, Apr 12 2020 *)
%t 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 *)
%o (PARI) t(n, k) = 30*k^2*(n-k)^2+1
%o trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
%o /* Print initial 9 rows of triangle as follows */ trianglerows(9)
%o (GAP) T:=Flat(List([0..9],n->List([0..n],k->30*k^2*(n-k)^2+1))); # _Muniru A Asiru_, Oct 24 2018
%o (Magma) [[30*k^2*(n-k)^2+1: k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Dec 14 2018
%o (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
%Y Various cases of L(m, n, k): A287326(m=1), This sequence (m=2), A300785(m=3). See comments for L(m, n, k).
%Y Row sums give the nonzero terms of A002561.
%Y Cf. A000584, A287326, A007318, A077028, A294317, A068236, A302971, A304042, A002561, A258807, A158558, A094053, A024003, A316349.
%K nonn,tabl,easy
%O 0,5
%A _Kolosov Petro_, Mar 10 2018