login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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