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.

Showing 1-3 of 3 results.

A081277 Square array of unsigned coefficients of Chebyshev polynomials of the first kind.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 8, 4, 1, 7, 18, 20, 8, 1, 9, 32, 56, 48, 16, 1, 11, 50, 120, 160, 112, 32, 1, 13, 72, 220, 400, 432, 256, 64, 1, 15, 98, 364, 840, 1232, 1120, 576, 128, 1, 17, 128, 560, 1568, 2912, 3584, 2816, 1280, 256, 1, 19, 162, 816, 2688, 6048, 9408, 9984, 6912
Offset: 0

Views

Author

Paul Barry, Mar 16 2003

Keywords

Comments

Rows include A011782, A001792, A001793, A001794, A006974.
Formatted as a triangular array, this is [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] (see construction in A084938 ). - Philippe Deléham, Aug 09 2005
Antidiagonal sums are in A025192. - Philippe Deléham, Dec 04 2006
Binomial transform of n-th row of the triangle (followed by zeros) = n-th row of the A142978 array and n-th column of triangle A104698. - Gary W. Adamson, Jul 17 2008
When formatted as a triangle, A038763=fusion of polynomial sequences (x+1)^n and (x+1)^n; see A193722 for the definition of fusion of two polynomial sequences or triangular arrays. Row n of A038763, as a triangle, consists of coefficients of the product (x+1)*(x+2)^n. - Clark Kimberling, Aug 04 2011

Examples

			Rows begin
  1, 1,  2,   4,   8, ...
  1, 3,  8,  20,  48, ...
  1, 5, 18,  56, 160, ...
  1, 7, 32, 120, 400, ...
  1, 9, 50, 220, 840, ...
  ...
As a triangle:
  1;
  1,  1;
  1,  3,  2;
  1,  5,  8,  4;
  1,  7, 18, 20,  8;

Links

Crossrefs

Cf. A079628.
Cf. A142978, A104698.
Cf. A167580 and A167591. - Johannes W. Meijer, Nov 23 2009
Cf. A053120 (antidiagonals give signed version) and A124182 (skewed version). - Mathias Zechmeister, Jul 26 2022

Programs

  • Mathematica
    (* Program generates triangle A081277 as the self-fusion of Pascal's triangle *)
z = 8; a = 1; b = 1; c = 1; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A081277 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* abs val of A118800 *)
Factor[w[6, x]]
(* Clark Kimberling, Aug 04 2011 *)

Formula

T(n, k) = (n+2k)*binomial(n+k-1, k-1)*2^(n-1)/k, k > 0.
T(n, 0) defined by g.f. (1-x)/(1-2x). Other rows are defined by (1-x)/(1-2x)^n.
T(n, 0) = 0 if n < 0, T(0, k) = 0 if k < 0, T(0, 0) = T(1, 0) = 1, T(n, k) = T(n, k-1) + 2*T(n-1, k); for example, 160 = 48 + 2*56 for n = 4 and k = 2. -Philippe Deléham, Aug 12 2005
G.f. of the triangular interpretation: (-1+x*y)/(-1+2*x*y+x). - R. J. Mathar, Aug 11 2015

A167591 A triangle related to the a(n) formulas of the rows of the ED4 array A167584.

Original entry on oeis.org

1, 4, -2, 12, -8, 9, 32, -16, 120, -60, 80, 0, 952, -768, 525, 192, 160, 5664, -5008, 12396, -5670, 448, 896, 27888, -20672, 162740, -133128, 72765, 1024, 3584, 120064, -46720, 1537216, -1562464, 2557296, -1081080, 2304, 12288, 467712, 76800
Offset: 1

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Comments

The a(n) formulas given below correspond to the first ten rows of the ED4 array A167584.
The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A013609, A003148, A081277 and A079628.

Examples

			Row 1: a(n) = 1.
Row 2: a(n) = 4*n - 2.
Row 3: a(n) = 12*n^2 - 8*n + 9.
Row 4: a(n) = 32*n^3 - 16*n^2 + 120*n - 60.
Row 5: a(n) = 80*n^4 + 0*n^3 + 952*n^2 - 768*n + 525.
Row 6: a(n) = 192*n^5 + 160*n^4 + 5664*n^3 - 5008*n^2 + 12396*n - 5670.
Row 7: a(n) = 448*n^6 + 896*n^5 + 27888*n^4 - 20672*n^3 + 162740*n^2 - 133128*n + 72765.
Row 8: a(n) = 1024*n^7 + 3584*n^6 + 120064*n^5 - 46720*n^4 + 1537216*n^3 - 1562464*n^2 + 2557296*n - 1081080.
Row 9: a(n) = 2304*n^8 + 12288*n^7 + 467712*n^6 + 76800*n^5 + 11589216*n^4 - 12058368*n^3 + 47963568*n^2 - 38278080*n + 18243225.
Row 10: a(n) = 5120*n^9 + 38400*n^8 + 1686528*n^7 + 1540608*n^6 + 73898880*n^5 - 66179520*n^4 + 631348672*n^3 - 669559008*n^2 + 869709780*n - 344594250.

Crossrefs

A167584 is the ED4 array.
A000012, A016825, A167585, A167586 and A167587 equal the first five rows of the ED4 array.
A001787, A167592, A167593, A168307 and A168308 equal the first five left hand triangle columns.
A001193 equals the first right hand triangle column.
A024199 equals the row sums.
Cf. A013609, A003148, A079628 and A081277.

Extensions

Comment and formulas added by Johannes W. Meijer, Nov 23 2009

A167580 A triangle related to the a(n) formulas of the rows of the ED3 array A167572.

Original entry on oeis.org

1, 6, -1, 20, 0, 3, 56, 28, 98, -15, 144, 192, 1080, -48, 105, 352, 880, 7568, 2024, 6534, -945, 832, 3328, 40976, 31616, 132444, -8112, 10395, 1920, 11200, 187488, 274480, 1593960, 286900, 972162, -135135, 4352, 34816, 761600, 1784320, 13962848
Offset: 1

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Comments

The a(n) formulas given below correspond to the first ten rows of the ED3 array A167572.
The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A013609, A003148, A081277 and A079628.

Examples

			Row 1: a(n) = 1.
Row 2: a(n) = 6*n - 1.
Row 3: a(n) = 20*n^2 + 0*n + 3.
Row 4: a(n) = 56*n^3 + 28*n^2 + 98*n - 15.
Row 5: a(n) = 144*n^4 + 192*n^3 + 1080*n^2 - 48*n + 105.
Row 6: a(n) = 352*n^5 + 880*n^4 + 7568*n^3 + 2024*n^2 + 6534*n - 945.
Row 7: a(n) = 832*n^6 + 3328*n^5 + 40976*n^4 + 31616*n^3 + 132444*n^2 - 8112*n + 10395.
Row 8: a(n) = 1920*n^7 + 11200*n^6 + 187488*n^5 + 274480*n^4 + 1593960*n^3 + 286900*n^2 + 972162*n - 135135.
Row 9: a(n) = 4352*n^8 + 34816*n^7 + 761600*n^6 + 1784320*n^5 + 13962848*n^4 + 7874944*n^3 + 29641200*n^2 - 2080800*n + 2027025.
Row 10: a(n) = 9728*n^9 + 102144*n^8 + 2830848*n^7 + 9645312*n^6 + 98382912*n^5 + 106720416*n^4 + 522283552*n^3 + 69265488*n^2 + 255468870*n - 34459425.

Crossrefs

A167572 is the ED3 array.
A000012, A016969, A167573, A167574 and A167575 equal the first five rows of the ED3 array.
A014480, A167581, A167582, A168305 and A168306 equal the first five left hand triangle columns.
A001147 equals the first right hand triangle column.
A167576 equals the row sums.
Cf. A013609, A003148, A079628 and A081277.

Extensions

Comment and links added by Johannes W. Meijer, Nov 23 2009
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.