A060102 -id:A060102 - cp's OEIS frontend

A060103 Fourth column (m=3) of triangle A060102.

1, 13, 71, 259, 742, 1806, 3906, 7722, 14223, 24739, 41041, 65429, 100828, 150892, 220116, 313956, 438957, 602889, 814891, 1085623, 1427426, 1854490, 2383030, 3031470, 3820635, 4773951, 5917653

Offset: 0

Wolfdieter Lang, Apr 06 2001

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A000012, A000290, A002417, A060102.

Table[((4n^2+20n+15)Binomial[n+4,4])/15,{n,0,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{1,13,71,259,742,1806,3906},30] (* Harvey P. Dale, Dec 16 2012 *)

a(n) = (4*n^2+20*n+15)*binomial(n+4, 4)/15.
G.f.: (1+6*x+x^2)/(1-x)^7.
a(0)=1, a(1)=13, a(2)=71, a(3)=259, a(4)=742, a(5)=1806, a(6)=3906, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Dec 16 2012

A060104 Fifth column (m=4) of triangle A060102.

Original entry on oeis.org

1, 19, 140, 660, 2370, 7062, 18348, 42900, 92235, 185185, 351208, 634712, 1100580, 1841100, 2984520, 4705464, 7237461, 10887855, 16055380, 23250700, 33120230, 46473570, 64314900, 87878700

Offset: 0

Wolfdieter Lang, Apr 06 2001

Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A060102, A060103.

Table[(2n^2+10n+7) Binomial[n+6,6]/7,{n,0,30}] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{1,19,140,660,2370,7062,18348,42900,92235},30] (* Harvey P. Dale, Nov 08 2012 *)

a(n) = (2*n^2+10*n+7)*binomial(n+6, 6)/7.
G.f.: (1+10*x+5*x^2)/(1-x)^9.
a(0)=1, a(1)=19, a(2)=140, a(3)=660, a(4)=2370, a(5)=7062, a(6)=18348, a(7)=42900, a(8)=92235, a(n)=9*a(n-1)-36*a(n-2)+84*a(n-3)- 126*a(n-4)+ 126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - Harvey P. Dale, Nov 08 2012

A060105 Sixth column (m=5) of triangle A060102.

Original entry on oeis.org

1, 26, 246, 1442, 6292, 22374, 68354, 185614, 458601, 1048476, 2246244, 4553484, 8801104, 16319252, 29174652, 50495236, 84906041, 139104966, 222612170, 348732670, 535778100, 808600650, 1200499950, 1755572130, 2531579505, 3603429336, 5067360936, 7045952056

Offset: 0

Wolfdieter Lang, Apr 06 2001

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A060104.

LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,26,246,1442,6292,22374,68354,185614,458601,1048476,2246244},30] (* Harvey P. Dale, Feb 22 2023 *)

a(n)=(n+4)*(8*n^2+64*n+45)*binomial(n+7, 7)/(4*45).

G.f.: (1+15*x+15*x^2+x^3)/(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)

A060557 Row sums of triangle A060556.

Original entry on oeis.org

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

A060556 Bisection of triangle A060098: odd-indexed members of column sequences of A060098 (not counting leading zeros).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 12, 16, 4, 1, 20, 50, 32, 5, 1, 30, 120, 140, 55, 6, 1, 42, 245, 448, 316, 86, 7, 1, 56, 448, 1176, 1284, 622, 126, 8, 1, 72, 756, 2688, 4170, 3102, 1113, 176, 9, 1, 90, 1200, 5544, 11550, 12122

Offset: 0

Wolfdieter Lang, Apr 06 2001

Row sums give A060557. Column sequences without leading zeros give for m=0..5: A000012 (powers of 1), A002378 = 2*A000217, A004320, 4*A040977, A060558, 2*A060559.
Companion triangle (even-indexed members) A060102.
With offset 1 for n and k, T(n,k) is the number of (1-2-3)-avoiding trapezoidal words of length n that contain n+1-k 1s. A trapezoidal word (following Riordan) is a sequence (a_1,a_2,...,a_n) of integers with 1 <= a_i <= 2i-1. For example, T(3,3)=3 counts 122, 132, 133 and T(4,2)=12 counts 1112, 1113, 1114, 1115, 1116, 1117, 1121, 1131, 1141, 1151, 1211, 1311. - David Callan, Aug 25 2009

			{1}; {1,2}; {1,6,3}; {1,12,16,4}; ...; Po(3,x) = 3 + x.

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

cp's OEIS Frontend

A060103 Fourth column (m=3) of triangle A060102.

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A060104 Fifth column (m=4) of triangle A060102.

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A060105 Sixth column (m=5) of triangle A060102.

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

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A060557 Row sums of triangle A060556.

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A060556 Bisection of triangle A060098: odd-indexed members of column sequences of A060098 (not counting leading zeros).

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