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A300656 Triangle read by rows: T(n,k) = 30*k^2*(n-k)^2+1; n >= 0, 0 <= k <= n. 9
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

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 = 2 in L(m,n,k) := Sum_{j=0..m} R(m,j) * k^j * (n-k)^j. For the cases m = 1 and m = 3 see A287326 and A300785, respectively.

For l = n, S1(m,l,n) = Sum_{k=1..l} Sum_{j=0..m} R(m,j) * 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(m,j) * 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

Kolosov Petro, Rows n = 0..2078 of triangle, flattened.

Petro Kolosov, Another Power Identity involving Binomial Theorem and Faulhaber's formula, arXiv:1603.02468 [math.NT], 2018.

FORMULA

T(n,k) = 30*k^2*(n-k)^2 + 1.

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).

From Kolosov Petro, Oct 30 2018: (Start)

Sum_{k=0..n} T(n,k) = A002561(n).

Sum_{k=1..n-1} T(n,k) = A258807(n). (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

T(n,k) = A158558((n-k)*k). - Kolosov Petro, Dec 07 2018

From Kolosov Petro, Dec 13 2018: (Start)

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.

T(n,k) = 30*A094053(n,k)^2 + 0*A094053(n,k)^1 + 1*A094053(n,k)^0.

Sum_{k=1..n-1} T(n,k) = -A024003(n), n > 1. (End)

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

(* 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 = 2 for A300656 *)

m=2;

(* 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 22 2018 *)

(* alternative program *)

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

Sequences A287326, A300785 represent the cases for m = 1, 3 in L(m,n,k), see comments, line 2.

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 October 16 21:10 EDT 2019. Contains 328103 sequences. (Running on oeis4.)