A059259 -id:A059259 - cp's OEIS frontend

A349839 Triangle T(n,k) built by placing all ones on the left edge, [1,0,0,0] repeated on the right edge, and filling the body using the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 1, 1, 4, 6, 4, 2, 0, 1, 5, 10, 10, 6, 2, 0, 1, 6, 15, 20, 16, 8, 2, 0, 1, 7, 21, 35, 36, 24, 10, 2, 1, 1, 8, 28, 56, 71, 60, 34, 12, 3, 0, 1, 9, 36, 84, 127, 131, 94, 46, 15, 3, 0, 1, 10, 45, 120, 211, 258, 225, 140, 61, 18, 3, 0, 1, 11, 55, 165, 331, 469, 483, 365, 201, 79, 21, 3, 1

Offset: 0

Michael A. Allen, Dec 01 2021

This is the m=4 member in the sequence of triangles A007318, A059259, A118923, A349839, A349841 which have all ones on the left side, ones separated by m-1 zeros on the other side, and whose interiors obey Pascal's recurrence.
T(n,k) is the (n,n-k)-th entry of the (1/(1-x^4),x/(1-x)) Riordan array.
For n>0, T(n,n-1) = A008621(n-1).
For n>1, T(n,n-2) = A001972(n-2).
For n>2, T(n,n-3) = A122046(n).
Sums of rows give A115451.
Sums of antidiagonals give A349840.

			Triangle begins:
  1;
  1,   0;
  1,   1,   0;
  1,   2,   1,   0;
  1,   3,   3,   1,   1;
  1,   4,   6,   4,   2,   0;
  1,   5,  10,  10,   6,   2,   0;
  1,   6,  15,  20,  16,   8,   2,   0;
  1,   7,  21,  35,  36,  24,  10,   2,   1;
  1,   8,  28,  56,  71,  60,  34,  12,   3,   0;
  1,   9,  36,  84, 127, 131,  94,  46,  15,   3,   0;
  1,  10,  45, 120, 211, 258, 225, 140,  61,  18,   3,   0;
  1,  11,  55, 165, 331, 469, 483, 365, 201,  79,  21,   3,   1;

Michael A. Allen and Kenneth Edwards, On Two Families of Generalizations of Pascal's Triangle, J. Int. Seq. 25 (2022) Article 22.7.1.

Other members of sequence of triangles: A007318, A059259, A118923, A349841.
Columns: A000012, A001477, A000217, A000292, A145126, A051745.
Diagonals: A121262, A008621, A001972, A122046.

Flatten[Table[CoefficientList[Series[(1-x*y)/((1-(x*y)^4)(1 - x - x*y)), {x, 0, 24}, {y, 0, 12}], {x, y}][[n+1,k+1]],{n,0,12},{k,0,n}]]

G.f.: (1-x*y)/((1-(x*y)^4)(1-x-x*y)) in the sense that T(n,k) is the coefficient of x^n*y^k in the series expansion of the g.f.
T(n,0) = 1.
T(n,n) = delta(n mod 4,0).
T(n,1) = n-1 for n>0.
T(n,2) = (n-1)*(n-2)/2 for n>1.
T(n,3) = (n-1)*(n-2)*(n-3)/6 for n>2.
T(n,4) = C(n-1,4) + 1 for n>3.
T(n,5) = C(n-1,5) + n - 5 for n>4.
For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/4)} binomial(n-4*j,n-k)/(n-4*j).
The g.f. of the n-th subdiagonal is 1/((1-x^4)(1-x)^n).

A030441 Values of Newton-Gregory forward interpolating polynomial (1/3)(2n - 3)(2n^2 - 3*n + 4).

Original entry on oeis.org

-4, -1, 2, 13, 40, 91, 174, 297, 468, 695, 986, 1349, 1792, 2323, 2950, 3681, 4524, 5487, 6578, 7805, 9176, 10699, 12382, 14233, 16260, 18471, 20874, 23477, 26288, 29315, 32566, 36049, 39772, 43743, 47970, 52461, 57224, 62267, 67598, 73225, 79156, 85399

Offset: 0

Ilias.Kotsireas(AT)lip6.fr (Ilias Kotsireas)

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Equals A030434 shifted left twice.

LinearRecurrence[{4,-6,4,-1},{-4,-1,2,13},50] (* Harvey P. Dale, Apr 20 2015 *)

a(n) = (1/3)*(2*n-3)*(2*n^2-3*n+4); \\ Michel Marcus, May 18 2014

a(n) - A177342(n-1) = (n-1)^2, with n>1. For n=6, a(6) - A177342(5) = 174 - 149 = 5^2. - Bruno Berselli, May 23 2010
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Colin Barker, May 18 2014
G.f.: (15*x^3-18*x^2+15*x-4) / (x-1)^4. - Colin Barker, May 18 2014
a(n) = A059259(2*n,3), n>1. - Mathew Englander, May 17 2014

A030442 Values of Newton-Gregory forward interpolating polynomial (1/6)(4n^4 - 60n^3 + 347n^2 - 927*n + 978).

Original entry on oeis.org

163, 57, 16, 4, 1, 3, 22, 86, 239, 541, 1068, 1912, 3181, 4999, 7506, 10858, 15227, 20801, 27784, 36396, 46873, 59467, 74446, 92094, 112711, 136613, 164132, 195616, 231429, 271951, 317578, 368722, 425811, 489289, 559616, 637268, 722737, 816531, 919174

Offset: 0

Ilias.Kotsireas(AT)lip6.fr (Ilias Kotsireas), Dec 11 1999

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A059259.

A030442:=n->(1/6)*(4*n^4-60*n^3+347*n^2-927*n+978); seq(A030442(n), n=0..40); # Wesley Ivan Hurt, May 19 2014

Table[(1/6)*(4*n^4 - 60*n^3 + 347*n^2 - 927*n + 978), {n, 0, 40}] (* Wesley Ivan Hurt, May 19 2014 *)

a(n) = (1/6)*(4*n^4-60*n^3+347*n^2-927*n+978); \\ Michel Marcus, May 18 2014

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Colin Barker, May 18 2014
G.f.: -(386*x^4-1136*x^3+1361*x^2-758*x+163) / (x-1)^5. - Colin Barker, May 18 2014
a(n) = A059259(2*n-5,4), n>4. - Mathew Englander, May 18 2014
E.g.f.: exp(x)*(978 - 636*x + 195*x^2 - 36*x^3 + 4*x^4)/6. - Stefano Spezia, Sep 11 2022

A180153 a(n) = 10*a(n-1) + A109242(n).

Original entry on oeis.org

1, 121, 12421, 1246421, 124696421, 12470296421, 1247037296421, 124703817296421, 12470382717296421, 1247038282717296421, 124703828392717296421, 12470382840592717296421, 1247038284073592717296421, 124703828407513592717296421, 12470382840753013592717296421

Offset: 1

Mark Dols, Aug 13 2010

Colin Barker, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (121,-2220,12100,-10000).

Cf. A109242, A000292, A002620, A059259, A094028.

Vec(x / ((x-1)*(10*x-1)^2*(100*x-1)) + O(x^30)) \\ Colin Barker, Oct 03 2015

From Colin Barker, Oct 03 2015: (Start)
a(n) = 121*a(n-1)-2220*a(n-2)+12100*a(n-3)-10000*a(n-4) for n>4.
G.f.: x / ((x-1)*(10*x-1)^2*(100*x-1)).
(End)

cp's OEIS Frontend

A349839 Triangle T(n,k) built by placing all ones on the left edge, [1,0,0,0] repeated on the right edge, and filling the body using the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

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A030441 Values of Newton-Gregory forward interpolating polynomial (1/3)*(2*n - 3)*(2*n^2 - 3*n + 4).

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A030442 Values of Newton-Gregory forward interpolating polynomial (1/6)*(4*n^4 - 60*n^3 + 347*n^2 - 927*n + 978).

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A180153 a(n) = 10*a(n-1) + A109242(n).

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A030441 Values of Newton-Gregory forward interpolating polynomial (1/3)(2n - 3)(2n^2 - 3*n + 4).

A030442 Values of Newton-Gregory forward interpolating polynomial (1/6)(4n^4 - 60n^3 + 347n^2 - 927*n + 978).