A060556 -id:A060556 - cp's OEIS frontend

A060557 Row sums of triangle A060556.

1, 3, 10, 33, 108, 352, 1145, 3721, 12087, 39254, 127469, 413908, 1343980, 4363921, 14169633, 46008619, 149389218, 485064009, 1574993356, 5113971944, 16604963593, 53915979657, 175064088671

Offset: 0

Wolfdieter Lang, Apr 06 2001

Equals the INVERT transform of A045623: (1, 2, 5, 12, 28, ...). - Gary W. Adamson, Oct 26 2010

Harry J. Smith, Table of n, a(n) for n = 0..500
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (5, -6, 1).

a(n)=A028495(2n+1).
Cf. A053975.
Cf. A052975 (row sums of triangle A060102).
Cf. A045623. - Gary W. Adamson, Oct 26 2010

a[0] = 1; a[1] = 3; a[2] = 10; a[n_] := a[n] = 5*a[n-1] - 6*a[n-2] + a[n-3]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jul 05 2013, after Floor van Lamoen *)
LinearRecurrence[{5,-6,1},{1,3,10},30] (* Harvey P. Dale, Nov 29 2013 *)

{ f="b060557.txt"; a0=1; a1=3; a2=10; write(f, "0 1"); write(f, "1 3"); write(f, "2 10"); for (n=3, 500, write(f, n, " ", a=5*a2 - 6*a1 + a0); a0=a1; a1=a2; a2=a; ) } \\ Harry J. Smith, Jul 07 2009

a(n) = Sum_{m=0..n} A060556(n, m).
G.f.: (1-x)^2/(1 - 5*x + 6*x^2 - x^3).
a(n) = 5a(n-1) - 6a(n-2) + a(n-3). - Floor van Lamoen, Nov 02 2005

A060558 Fifth column (m=4) of triangle A060556.

Original entry on oeis.org

5, 55, 316, 1284, 4170, 11550, 28380, 63492, 131703, 256685, 474760, 839800, 1429428, 2352732, 3759720, 5852760, 8900265, 13252899, 19362596, 27804700, 39303550, 54761850, 75294180, 102265020

Offset: 0

Wolfdieter Lang, Apr 06 2001

Harry J. Smith, Table of n, a(n) for n = 0..500

Cf. A000217, A004320, A040977.

a(n)={(2*n^2 + 18*n + 35)*binomial(n + 6, 6)/7} \\ Harry J. Smith, Jul 07 2009

a(n) = (2*n^2 + 18*n + 35)*binomial(n+6, 6)/7.

G.f.: (5 + 10*x + x^2)/(1-x)^9.

A060559 One half of sixth column (m=5) of triangle A060556.

Original entry on oeis.org

3, 43, 311, 1551, 6061, 19877, 57057, 147433, 349778, 773058, 1609322, 3182842, 6021366, 10956806, 19264366, 32851038, 54506573, 88232485, 139667385, 216629985, 329804475, 493596675, 727193415, 1055862015

Offset: 0

Wolfdieter Lang, Apr 06 2001

Harry J. Smith, Table of n, a(n) for n = 0..500

Cf. A060558.

a(n)={(8*n^2 + 72*n + 135)*binomial(n + 8, 8)/45} \\ Harry J. Smith, Jul 07 2009

a(n) = (8*n^2 + 72*n + 135)*binomial(n+8, 8)/45.

G.f.: (3 + 10*x + 3*x^2)/(1-x)^11.

A060098 Triangle of partial sums of column sequences of triangle A060086, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 9, 16, 13, 5, 1, 1, 12, 30, 32, 19, 6, 1, 1, 16, 50, 71, 55, 26, 7, 1, 1, 20, 80, 140, 140, 86, 34, 8, 1, 1, 25, 120, 259, 316, 246, 126, 43, 9, 1, 1, 30, 175, 448, 660, 622, 399, 176, 53, 10, 1

Offset: 0

Wolfdieter Lang, Apr 06 2001

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The g.f. for the row polynomials p(n,x) = Sum_{m=0..n} a(n,m)*x^m is (1/(1-x*z/((1-z^2)*(1-z))))/(1-z).
Row sums give A052534. Column sequences (without leading zeros) give A000012 (powers of 1), A002620(n+1), A002624, A060099-A060101 for m=0..5.
The bisections of the column sequences give triangles A060102 and A060556.
Riordan array (1/(1-x), x/((1-x)*(1-x^2))). - Paul Barry, Mar 28 2011

			p(3,x) = 1 + 4*x + 3*x^2 + x^3.
Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  3,  1;
  1,  6,  8,  4,  1;
  1,  9, 16, 13,  5,  1;
  1, 12, 30, 32, 19,  6,  1;
  1, 16, 50, 71, 55, 26,  7,  1;
  ...

Vincenzo Librandi, Rows n = 0..100, flattened
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.

Cf. A052534, A000012, A002620(n+1), A002624, A060099, A060100, A060101.

Cf. A060102, A060556, A188316.

A060098 := proc(n,k) add( binomial(n-2*i,n-2*i-k)*binomial(k+i-1,i), i=0..floor(n/2)) ; end proc:
seq(seq(A060098(n,k), k=0..n), n=0..12); # R. J. Mathar, Mar 29 2011
# Recurrence after Philippe Deléham:
T := proc(n, k) option remember;
if k < 0 or k > n then 0 elif k = 0 or n = k then 1 else
T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, May 07 2023

t[n_, k_] := Sum[ Binomial[n-2*j, n-2*j-k]*Binomial[k+j-1, j], {j, 0, n/2}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

G.f. for column m >= 0: ((x/((1-x^2)*(1-x)))^m)/(1-x) = x^m/((1+x)^m*(1-x)^(2*m+1)).
Number triangle T(n,k) = Sum_{i=0..floor(n/2)} C(n-2*i,n-2*i-k)*C(k+i-1,i). - Paul Barry, Mar 28 2011
From Philippe Deléham, Apr 20 2023: (Start)
T(n, k) = 0 if k < 0 or if k > n; T(n, k) = 1 if k = 0 or k = n; otherwise:
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k).
T(n, k) = A188316(n, k) + A188316(n-1, k). (End)

A060102 Bisection of triangle A060098: even-indexed members of column sequences of A060098 (not counting leading zeros).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 8, 1, 1, 16, 30, 13, 1, 1, 25, 80, 71, 19, 1, 1, 36, 175, 259, 140, 26, 1, 1, 49, 336, 742, 660, 246, 34, 1, 1, 64, 588, 1806, 2370, 1442, 399, 43, 1, 1, 81, 960, 3906, 7062, 6292, 2828, 610, 53

Offset: 0

Wolfdieter Lang, Apr 06 2001

Row sums give A052975. Column sequences without leading zeros give for m=0..5: A000012 (powers of 1), A000290 (squares), A002417(n+1), A060103-5.

Companion triangle (odd-indexed members) A060556.

			{1}; {1,1}; {1,4,1}; {1,9,8,1}; ... Pe(3,x) = 1 + 3*x.

a(n, m) = A060098(2*n-m, m).
a(n, m) = Sum_{j=0..floor((m+1)/2)} binomial((n-m)-j+2*m, 2*m)*binomial(m+1, 2*j), n >= m >= 0, otherwise zero.
G.f. for column m: (x^m)*Pe(m+1, x)/(1-x)^(2*m+1), with Pe(n, x) = Sum_{j=0..floor(n/2)} binomial(n, 2*j)*x^j (even members of row n of Pascal triangle A007318).

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A060557 Row sums of triangle A060556.

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A060558 Fifth column (m=4) of triangle A060556.

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A060559 One half of sixth column (m=5) of triangle A060556.

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A060098 Triangle of partial sums of column sequences of triangle A060086, read by rows.

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A060102 Bisection of triangle A060098: even-indexed members of column sequences of A060098 (not counting leading zeros).

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