A133819 -id:A133819 - cp's OEIS frontend

A133821 Triangle whose rows are sequences of increasing fourth powers: 1; 1,16; 1,16,81; ... .

1, 1, 16, 1, 16, 81, 1, 16, 81, 256, 1, 16, 81, 256, 625, 1, 16, 81, 256, 625, 1296, 1, 16, 81, 256, 625, 1296, 2401, 1, 16, 81, 256, 625, 1296, 2401, 4096, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000

Offset: 1

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,16,1,16,81,1,16,81,256,..., analogous to A002260.

			Triangle starts
1;
1, 16;
1, 16; 81;
1, 16, 81, 256;
1, 16, 81, 256, 625;

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A000538 (row sums), A002260, A133819, A133820, A133824.

a133821 n k = a133821_tabl !! (n-1) !! (k-1)
a133821_row n = a133821_tabl !! (n-1)
a133821_tabl = map (`take` (tail a000583_list)) [1..]
-- Reinhard Zumkeller, Nov 11 2012

Module[{nn=10,fp},fp=Range[(nn(nn+1))/2]^4;Table[TakeList[fp,{n}],{n,nn}]]//Flatten (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Mar 29 2020 *)

O.g.f.: (1+11qx+11q^2x^2+q^3x^3)/((1-x)(1-qx)^5) = 1 + x(1 + 16q) + x^2(1 + 16q + 81q^2) + ... . Cf. 4th row of A008292.

Offset changed by Reinhard Zumkeller, Nov 11 2012

A143814 Triangle T(n,m) read along rows: T(n,m) = n^2 - (m+1)^2 for 1<=m

Original entry on oeis.org

3, 5, 8, 12, 7, 15, 21, 16, 9, 24, 32, 27, 20, 11, 35, 45, 40, 33, 24, 13, 48, 60, 55, 48, 39, 28, 15, 63, 77, 72, 65, 56, 45, 32, 17, 80, 96, 91, 84, 75, 64, 51, 36, 19, 99, 117, 112, 105, 96, 85, 72, 57, 40, 21, 120, 140, 135, 128, 119, 108, 95, 80, 63, 44, 23, 143

Offset: 2

Paul Curtz, Sep 02 2008

The triangle appears taking the entries of A140978,
4;
9,9;
16,16,16;
25,25,25,25;
..
minus the entries of A133819 with the 1's moved to the end of the rows,
1;
4,1;
4,9,1;
4,9,16,1;
4,9,16,25,1;
The result T(n,m) is a variant of A120070, the first term in each row of A120070 transferred to the end of the row.

			3;
5,8;
12,7,15;
21,16,9,24;
32,27,20,11,35;

Cf. A016061 (row sums).

A143814 := proc(n,m) if mA143814(n,m),m=1..n-1),n=2..14) ; # R. J. Mathar, Jan 23 2011

A143844 Triangle T(n,k) = k^2 read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 4, 0, 1, 4, 9, 0, 1, 4, 9, 16, 0, 1, 4, 9, 16, 25, 0, 1, 4, 9, 16, 25, 36, 0, 1, 4, 9, 16, 25, 36, 49, 0, 1, 4, 9, 16, 25, 36, 49, 64, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121

Offset: 0

Paul Curtz, Sep 03 2008

This is triangle A133819 with an additional leading column of zeros.

There is a family of even integer-valued polynomials p_n(x) = product_{k=0..n} (x^2 - T(n,k))/ A002674(n+1). We find p_0(x) in A000290, p_1(x) in A002415, p_2(x) essentially in A040977, p_3(x) in A053347 and p_4(x) in A054334. - Paul Curtz, Jun 10 2011

Cf. A000290, A002262, A002415, A002674, A040977, A053347, A054334, A133819.

Table[Range[0,n]^2,{n,0,15}]//Flatten (* Harvey P. Dale, Sep 08 2017 *)

for(n=0,9,for(k=0,n,print1(k^2", "))) \\ Charles R Greathouse IV, Jun 10 2011

T(n,k) = (A002262(n,k))^2.

G.f.: x*y*(1 + x*y)/((1 - x)*(1 - x*y)^3). - Stefano Spezia, Feb 21 2024

Definition simplified by R. J. Mathar, Sep 07 2009

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A133821 Triangle whose rows are sequences of increasing fourth powers: 1; 1,16; 1,16,81; ... .

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A143814 Triangle T(n,m) read along rows: T(n,m) = n^2 - (m+1)^2 for 1<=m

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A143844 Triangle T(n,k) = k^2 read by rows.

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