cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A347056 Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.

Original entry on oeis.org

1, 3, 4, 6, 16, 10, 10, 40, 50, 20, 15, 80, 150, 120, 35, 21, 140, 350, 420, 245, 56, 28, 224, 700, 1120, 980, 448, 84, 36, 336, 1260, 2520, 2940, 2016, 756, 120, 45, 480, 2100, 5040, 7350, 6720, 3780, 1200, 165, 55, 660, 3300, 9240, 16170, 18480, 13860, 6600, 1815, 220
Offset: 0

Views

Author

Luc Rousseau, Aug 14 2021

Keywords

Comments

This triangle is T[3] in the sequence (T[p]) of triangles defined by: T[p](n,k) = (k+p)*(n+p-1)!/(k!*(n-k)!*p!) and T[0](0,0)=1.
Riordan triangle (1/(1-x)^3, x/(1-x)) with column k scaled with A000292(k+1) = binomial(k+3, 3), for k >= 0. - Wolfdieter Lang, Sep 30 2021

Examples

			T(6,2) = (6+1)*(6+2)*(2+3)*binomial(6,2)/6 = 7*8*5*15/6 = 700.
The triangle T begins:
n \ k  0   1    2     3     4     5     6     7     8    9  10 ...
0:     1
1:     3   4
2:     6  16   10
3:    10  40   50    20
4:    15  80  150   120    35
5:    21 140  350   420   245    56
6:    28 224  700  1120   980   448    84
7:    36 336 1260  2520  2940  2016   756   120
8:    45 480 2100  5040  7350  6720  3780  1200   165
9:    55 660 3300  9240 16170 18480 13860  6600  1815  220
10:   66 880 4950 15840 32340 44352 41580 26400 10890 2640 286
... - _Wolfdieter Lang_, Sep 30 2021

Links

Crossrefs

Cf. A097805 (p=0), A103406 (p=1), A124932 (essentially p=2).
From Wolfdieter Lang, Sep 30 2021: (Start)
Columns (with leading zeros): A000217(n+1), 4*A000294, 10*A000332(n+2), 20*A000389(n+2), 35*A000579(n+2), 56*A000580(n+2), 84*A000581(n+2), 120*A000582(n+2), ...
Diagonals: A000292(k+1), A004320(k+1), 2*A006411(k+1), 10*A040977, ... (End)

Programs

  • PARI
    T(p,n,k)=if(n==0&&p==0,1,((k+p)*(n+p-1)!)/(k!*(n-k)!*p!))
for(n=0,9,for(k=0,n,print1(T(3,n,k),", ")))

Formula

T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6.
G.f. column k: x^k*binomial(k+3, 3)/(1 - x)^(k+3), for k >= 0. - Wolfdieter Lang, Sep 30 2021

A177815 Triangle read by rows: binomial(n, m^3).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 1, 9, 9, 1, 10, 45, 1, 11, 165, 1, 12, 495, 1, 13, 1287, 1, 14, 3003, 1, 15, 6435, 1, 16, 12870, 1, 17, 24310, 1, 18, 43758, 1, 19, 75582, 1, 20, 125970
Offset: 0

Views

Author

Roger L. Bagula, Dec 13 2010

Keywords

Comments

Row sums are: {1, 2, 3, 4, 5, 6, 7, 8, 10, 19, 56, 177, 508, 1301, 3018, 6451, 12887, 24328, 43777, 75602, 125991,...}.
First length 4 row is: {1, 27, 2220075, 1}.

Examples

			{1},
{1, 1},
{1, 2},
{1, 3},
{1, 4},
{1, 5},
{1, 6},
{1, 7},
{1, 8, 1},
{1, 9, 9},
{1, 10, 45},
{1, 11, 165},
{1, 12, 495},
{1, 13, 1287},
{1, 14, 3003},
{1, 15, 6435},
{1, 16, 12870},
{1, 17, 24310},
{1, 18, 43758},
{1, 19, 75582},
{1, 20, 125970},
...
{1, 27, 2220075, 1}

Crossrefs

Cf. A003099, A181543, A000581.

Programs

  • Mathematica
    t[n_, m_] = Binomial[n, m^3];
Table[Table[t[n, m], {m, 0, Floor[n^(1/3)]}], {n, 0, 20}];
Flatten[%]

A356037 Conjecturally, a(n) is the smallest number m such that every natural number is a sum of at most m n-simplex numbers.

Original entry on oeis.org

1, 3, 5, 8, 10, 13, 15, 15, 19, 24
Offset: 1

Views

Author

Mohammed Yaseen, Jul 24 2022

Keywords

Comments

n-simplex numbers are {binomial(k,n); k>=n}.
This problem is the simplex number analog of Waring's problem.
a(2) = 3 was proposed by Fermat and proved by Gauss, see A061336.
Pollock conjectures that a(3) = 5. Salzer and Levine prove this for numbers up to 452479659. See A104246 and A000797.
Kim gives a(4)=8, a(5)=10, a(6)=13 and a(7)=15 (not proved).

Examples

			2-simplex numbers are {binomial(k,2); k>=2} = {1,3,6,10,...}, the triangular numbers. 3 is the smallest number m such that every natural number is a sum of at most m triangular numbers. So a(2)=3.
3-simplex numbers are {binomial(k,3); k>=3} = {1,4,10,20,...}, the tetrahedral numbers. 5 is presumed to be the smallest number m such that every natural number is a sum of at most m tetrahedral numbers. So a(3)=5.

Links

Crossrefs

Cf. A002804, A079611.
Minimal number of x-simplex numbers whose sum equals n: A061336 (x=2), A104246 (x=3), A283365 (x=4), A283370 (x=5).
x-simplex numbers: A000217 (x=2), A000292 (x=3), A000332 (x=4), A000389 (x=5), A000579 (x=6), A000580 (x=7), A000581 (x=8), A000582 (x=9).
Previous Showing 51-53 of 53 results.

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