A113139 -id:A113139 - cp's OEIS frontend

A264772 Triangle T(n,k) = binomial(3n - 2k, 2*n - k), 0 <= k <= n.

1, 3, 1, 15, 4, 1, 84, 21, 5, 1, 495, 120, 28, 6, 1, 3003, 715, 165, 36, 7, 1, 18564, 4368, 1001, 220, 45, 8, 1, 116280, 27132, 6188, 1365, 286, 55, 9, 1, 735471, 170544, 38760, 8568, 1820, 364, 66, 10, 1, 4686825, 1081575, 245157, 54264, 11628, 2380, 455, 78, 11, 1

Offset: 0

Peter Bala, Nov 24 2015

Riordan array (f(x), x*g(x)), where g(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + ... is the o.g.f. for A001764 and f(x) = g(x)/(3 - 2*g(x)) = 1 + 3*x + 15*x^2 + 84*x^3 + 495*x^4 + ... is the o.g.f. for A005809.
The even numbered columns give the Riordan array A119301, the odd numbered columns give the Riordan array A144484. A159841 is the array formed from columns 1,4,7,10,....
More generally, if R = (R(n,k))n,k>=0 is a proper Riordan array, m is a nonnegative integer and a > b are integers then the array with (n,k)-th element R((m + 1)*n - a*k, m*n - b*k) is also a Riordan array (not necessarily proper). Here we take R as Pascal's triangle and m = a = 2, b = 1. See A092392, A264773, A264774 and A113139 for further examples.

			Triangle begins
.n\k.|......0.....1....2....3...4..5...6..7...
----------------------------------------------
..0..|      1
..1..|      3     1
..2..|     15     4    1
..3..|     84    21    5    1
..4..|    495   120   28    6   1
..5..|   3003   715  165   36   7  1
..6..|  18564  4368 1001  220  45  8  1
..7..| 116280 27132 6188 1365 286 55  9  1
...

Michael De Vlieger, Table of n, a(n) for n = 0..11475
Peter Bala, A 4-parameter family of embedded Riordan arrays
Paul Barry, On the halves of a Riordan array and their antecedents, arXiv:1906.06373 [math.CO], 2019.
E. Lebensztayn, On the asymptotic enumeration of accessible automata, Discrete Mathematics and Theoretical Computer Science, Vol. 12, No.3, 2010, 75-80, Section 2.
R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics, CreateSpace Independent Publishing Platform 2006, Section 5.6, ISBN-13: 978-1502925244.

Cf. A005809 (column 0), A045721 (column 1), A025174 (column 2), A004319 (column 3), A236194 (column 4), A013698 (column 5). Cf. A001764, A007318, A092392, A119301 (C(3n-k,2n)), A144484 (C(3n+1-k,2n+1)), A159841 (C(3n+1,2n+k+1)), A264773, A264774.

/* As triangle */ [[Binomial(3*n-2*k, n-k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Dec 02 2015

A264772:= proc(n,k) binomial(3*n - 2*k, 2*n - k); end proc:
seq(seq(A264772(n,k), k = 0..n), n = 0..10);

Table[Binomial[3 n - 2 k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 01 2015 *)

T(n,k) = binomial(3*n - 2*k, n - k).

O.g.f.: f(x)/(1 - t*x*g(x)), where f(x) = Sum_{n >= 0} binomial(3*n,n)*x^n and g(x) = Sum_{n >= 0} 1/(2*n + 1)*binomial(3*n,n)*x^n.

A190666 Number of walks from (0,0) to (n+3,n) which take steps from {E, N, NE}.

Original entry on oeis.org

1, 9, 61, 377, 2241, 13073, 75517, 433905, 2485825, 14218905, 81270333, 464387817, 2653649025, 15167050785, 86716873725, 495998874593, 2838240338817, 16248650965289, 93065296937533, 533285164334169, 3057236753252161, 17534423944871729, 100609937775369981

Offset: 0

Shanzhen Gao, May 25 2011

+-3-diagonal of A008288 as a square array. - Shel Kaphan, Jan 07 2023

S. Gao, H. Niederhausen, Counting New Lattice Paths and Walks with Several Step Vectors (submitted to Congr. Numer.). - Shanzhen Gao, May 25 2011

Alois P. Heinz, Table of n, a(n) for n = 0..400
Shanzhen Gao and Keh-Hsun Chen, Counting Lattice Paths and Walks with Several Step Vectors, FCS 2014.

Cf. A001850, A008288, A113139, A002002, A026002.

b:= proc(i, j) option remember;
      if i<0 or j<0 then 0
    elif i=0 and j=0 then 1
    else b(i-1, j) +b(i, j-1) +b(i-1, j-1)
      fi
    end:
a:= n-> b(n+3, n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 28 2011

b[i_, j_] /; i<0 || j<0 = 0; b[0, 0] = 1; b[i_, j_]:= b[i, j]= b[i-1, j] + b[i, j-1] + b[i-1, j-1]; a[n_] := b[n+3, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 01 2011, after Maple prog. *)
CoefficientList[Series[(-1+3*x-x^2+(1-6*x+6*x^2-x^3)/Sqrt[x^2-6*x+1])/(2*x^3), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
Table[(-1)^n Hypergeometric2F1[-n, n+4, 1, 2], {n,0,22}] (* Peter Luschny, Mar 02 2017 *)

a(n) = Sum_{k=0..n} C(n,k) * C(n+k+3,k+3) = A113139 (n+3,3). - Alois P. Heinz, Jun 01 2011
G.f.: (-1 + 3*x - x^2 + (1 - 6*x + 6*x^2 - x^3)/sqrt(x^2 - 6*x + 1))/(2*x^3). - Alois P. Heinz, Jun 03 2011
Recurrence: n*(n+3)*a(n) = (5*n^2 + 15*n + 16)*a(n-1) + (5*n^2 - 5*n + 6)*a(n-2) - (n-2)*(n+1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ sqrt(1632 + 1154*sqrt(2))*(3 + 2*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012
From Peter Bala, Mar 02 2017: (Start)
a(n) = (1/2^(n+1))*Sum_{k >= 3} (1/2^k)*binomial(n+k, k)*binomial(n+k, n+3).
a(n) = (-1)^n*Sum_{k = 0..n} (-2)^k*binomial(n,k) * binomial(n+k+3,k).
n*(n+3)*(2*n + 1)*a(n) = 6*(n+1)*(2*n^2 + 4*n + 3)*a(n-1) - (n-1)*(n+2)*(2*n + 3)*a(n-2) with a(0) = 1 and a(1) = 9. (End)
a(n) = (-1)^n*hypergeom([-n, n+4], [1], 2). - Peter Luschny, Mar 02 2017

A110171 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n that start with exactly k (0,1) steps (or, equivalently, with exactly k (1,0) steps).

Original entry on oeis.org

1, 2, 1, 8, 4, 1, 38, 18, 6, 1, 192, 88, 32, 8, 1, 1002, 450, 170, 50, 10, 1, 5336, 2364, 912, 292, 72, 12, 1, 28814, 12642, 4942, 1666, 462, 98, 14, 1, 157184, 68464, 27008, 9424, 2816, 688, 128, 16, 1, 864146, 374274, 148626, 53154, 16722, 4482, 978, 162, 18, 1

Offset: 0

Emeric Deutsch, Jul 14 2005

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).
Column k for k >= 1 has g.f. z^k*R^(k-1)*g*(1+z*R), where R = 1 + zR + zR^2 = (1 - z - sqrt(1-6z+z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318) and g = 1/sqrt(1-6z+z^2) is the g.f. of the central Delannoy numbers (A001850).
Sum_{k=0..n} k*T(n,k) = A050151(n) (the partial sums of the central Delannoy numbers) = (1/2)*n*R(n), where R(n) = A006318(n) is the n-th large Schroeder number.
From Paul Barry, May 07 2009: (Start)
Riordan array ((1+x+sqrt(1-6x+x^2))/(2*sqrt(1-6x+x^2)), (1-x-sqrt(1-6x+x^2))/2).
Inverse of Riordan array ((1-2x-x^2)/(1-x^2), x(1-x)/(1+x)). (End)

			T(2,1)=4 because we have NED, NENE, NEEN and NDE.
Triangle starts:
    1;
    2,  1;
    8,  4,  1;
   38, 18,  6,  1;
  192, 88, 32,  8,  1;
From _Paul Barry_, May 07 2009: (Start)
Production matrix is
   2, 1,
   4, 2, 1,
   6, 2, 2, 1,
   8, 2, 2, 2, 1,
  10, 2, 2, 2, 2, 1,
  12, 2, 2, 2, 2, 2, 1,
  14, 2, 2, 2, 2, 2, 2, 1,
  16, 2, 2, 2, 2, 2, 2, 2, 1,
  18, 2, 2, 2, 2, 2, 2, 2, 2, 1 (End)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
W.-j. Woan, The Lagrange Inversion Formula and Divisibility Properties, JIS 10 (2007) 07.7.8, example 5.

Cf. A001850, A002003, A006318, A050146, A050151, A113139, A118384.

Q:=sqrt(1-6*z+z^2): G:=(1+z+Q)/Q/(2-t+t*z+t*Q): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 10 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form

T[n_, n_] = 1;
T[n_, k_] := Sum[Binomial[n, i] Binomial[2n-k-i-1, n-k-i], {i, 0, n}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jun 13 2019 *)

A110171 = lambda n,k : binomial(n, k)*hypergeometric([k-n, n], [k+1], -1)
for n in (0..9): [round(A110171(n,k).n(100)) for k in (0..n)] # Peter Luschny, Sep 17 2014

T(n,0) = A002003(n) for n >= 1.
T(n,1) = A050146(n) for n >= 1.
Row sums are the central Delannoy numbers (A001850).
G.f.: (1+z+Q)/(Q(2-t+tz+tQ)), where Q=sqrt(1-6z+z^2).
T(n,k) = x^(n-k)*((1+x)/(1-x))^n. - Paul Barry, May 07 2009
T(n,k) = C(n, k)*hypergeometric([k-n, n], [k+1], -1). - Peter Luschny, Sep 17 2014
From Peter Bala, Jun 29 2015: (Start)
T(n,k) = Sum_{i = 0..n} binomial(n,i)*binomial(2*n-k-i-1,n-k-i).
Matrix product A118384 * A007318^(-1)
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = ( 1 - x - sqrt(1 - 6*x + x^2) )/2 and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan). (End)
T(n,k) = P(n-k, k, -1, 3), where P(n, alpha, beta, x) is the n-th Jacobi polynomial with parameters alpha and beta. Cf. A113139. - Peter Bala, Feb 16 2020

A264773 Triangle T(n,k) = binomial(4n - 3k, 3n - 2k), 0 <= k <= n.

Original entry on oeis.org

1, 4, 1, 28, 5, 1, 220, 36, 6, 1, 1820, 286, 45, 7, 1, 15504, 2380, 364, 55, 8, 1, 134596, 20349, 3060, 455, 66, 9, 1, 1184040, 177100, 26334, 3876, 560, 78, 10, 1, 10518300, 1560780, 230230, 33649, 4845, 680, 91, 11, 1, 94143280, 13884156, 2035800, 296010, 42504, 5985, 816, 105, 12, 1

Offset: 0

Peter Bala, Nov 30 2015

Riordan array (f(x),x*g(x)), where g(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + ... is the o.g.f. for A002293 and f(x) = g(x)/(4 - 3*g(x)) = 1 + 4*x + 28*x^2 + 220*x^3 + 1820*x^4 + ... is the o.g.f. for A005810.

More generally, if (R(n,k))n,k>=0 is a proper Riordan array and m is a nonnegative integer and a > b are integers then the array with (n,k)-th element R((m + 1)*n - a*k, m*n - b*k) is also a Riordan array (not necessarily proper). Here we take R as Pascal's triangle and m = a = 3 and b = 2. See A092392, A264772, A264774 and A113139 for further examples.

			Triangle begins
  n\k |       0      1     2    3   4   5   6   7
------+-----------------------------------------------
   0  |       1
   1  |       4      1
   2  |      28      5     1
   3  |     220     36     6    1
   4  |    1820    286    45    7   1
   5  |   15504   2380   364   55   8   1
   6  |  134596  20349  3060  455  66   9   1
   7  | 1184040 177100 26334 3876 560  78  10   1
...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).
Peter Bala, A 4-parameter family of embedded Riordan arrays
E. Lebensztayn, On the asymptotic enumeration of accessible automata, Discrete Mathematics and Theoretical Computer Science, Vol. 12, No. 3, 2010, 75-80, Section 2.
R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics CreateSpace Independent Publishing Platform 2006, Section 5.6, ISBN-13: 978-1502925244.

A005810 (column 0), A052203 (column 1), A257633 (column 2), A224274 (column 3), A004331 (column 4). Cf. A002293, A007318, A092392 (C(2n-k,n)), A119301 (C(3n-k,n-k)), A264772, A264774.

/* As triangle: */ [[Binomial(4*n-3*k, 3*n-2*k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Dec 02 2015

A264773:= proc(n,k) binomial(4*n - 3*k, 3*n - 2*k); end proc:
seq(seq(A264773(n,k), k = 0..n), n = 0..10);

A264773[n_,k_] := Binomial[4*n - 3*k, n - k];
Table[A264773[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 06 2024 *)

T(n,k) = binomial(4*n - 3*k, n - k).

O.g.f.: f(x)/(1 - t*x*g(x)), where f(x) = Sum_{n >= 0} binomial(4*n,n)*x^n and g(x) = Sum_{n >= 0} 1/(3*n + 1)*binomial(4*n,n)*x^n.

A264774 Triangle T(n,k) = binomial(5n - 4k, 4n - 3k), 0 <= k <= n.

Original entry on oeis.org

1, 5, 1, 45, 6, 1, 455, 55, 7, 1, 4845, 560, 66, 8, 1, 53130, 5985, 680, 78, 9, 1, 593775, 65780, 7315, 816, 91, 10, 1, 6724520, 736281, 80730, 8855, 969, 105, 11, 1, 76904685, 8347680, 906192, 98280, 10626, 1140, 120, 12, 1, 886163135, 95548245, 10295472, 1107568, 118755, 12650, 1330, 136, 13, 1

Offset: 0

Peter Bala, Nov 30 2015

Riordan array (f(x),x*g(x)), where g(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + ... is the o.g.f. for A002294 and f(x) = g(x)/(5 - 4*g(x)) = 1 + 5*x + 45*x^2 + 455*x^3 + 4845*x^4 + ... is the o.g.f. for A001449.

More generally, if (R(n,k))n,k>=0 is a proper Riordan array and m is a nonnegative integer and a > b are integers then the array with (n,k)-th element R((m + 1)*n - a*k, m*n - b*k) is also a Riordan array (not necessarily proper). Here we take R as Pascal's triangle and m = a = 4 and b = 3. See A092392, A264772, A264773 and A113139 for further examples.

			Triangle begins
  n\k |       0      1     2    3   4   5   6   7
------+---------------------------------------------
   0  |       1
   1  |       5      1
   2  |      45      6     1
   3  |     455     55     7    1
   4  |    4845    560    66    8   1
   5  |   53130   5985   680   78   9   1
   6  |  593775  65780  7315  816  91  10   1
   7  | 6724520 736281 80730 8855 969 105  11  1
...

Peter Bala, A 4-parameter family of embedded Riordan arrays
E. Lebensztayn, On the asymptotic enumeration of accessible automata, Section 2, Discrete Mathematics and Theoretical Computer Science, Vol. 12, No. 3, 2010, 75-80, Section 2.
R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics, CreateSpace Independent Publishing Platform 2006, Section 5.6, ISBN-13: 978-1502925244.

Cf. A001449 (column 0), A079589(column 1). Cf. A002294, A007318, A092392 (C(2n-k,n)), A113139, A119301 (C(3n-k,n-k)), A264772, A264773.

/* As triangle */ [[Binomial(5*n-4*k, 4*n-3*k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Dec 02 2015

A264774:= proc(n,k) binomial(5*n - 4*k, 4*n - 3*k); end proc:
seq(seq(A264774(n,k), k = 0..n), n = 0..10);

Table[Binomial[5 n - 4 k, 4 n - 3 k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 01 2015 *)

T(n,k) = binomial(5*n - 4*k, n - k).

O.g.f.: f(x)/(1 - t*x*g(x)), where f(x) = Sum_{n >= 0} binomial(5*n,n)*x^n and g(x) = Sum_{n >= 0} 1/(4*n + 1)*binomial(5*n,n)*x^n.

A113141 Inverse of a Delannoy related triangle.

Original entry on oeis.org

1, -3, 1, 2, -5, 1, -2, 10, -7, 1, 2, -14, 22, -9, 1, -2, 18, -46, 38, -11, 1, 2, -22, 78, -106, 58, -13, 1, -2, 26, -118, 230, -202, 82, -15, 1, 2, -30, 166, -426, 538, -342, 110, -17, 1, -2, 34, -222, 710, -1194, 1082, -534, 142, -19, 1, 2, -38, 286, -1098, 2330, -2814, 1958, -786, 178, -21, 1

Offset: 0

Paul Barry, Oct 15 2005

First column is (-1)^n*A104435(n). Row sums are 1,-2,-2,2,2,-2,-2,... Second column is A113142. Inverse of A113139.

			Triangle begins
1;
-3, 1;
2, -5, 1;
-2, 10, -7, 1;
2, -14, 22, -9, 1;
-2, 18, -46, 38, -11, 1;

A113140 Diagonal sums of a Delannoy related triangle.

Original entry on oeis.org

1, 3, 14, 68, 347, 1819, 9712, 52532, 286905, 1578679, 8738278, 48601968, 271406347, 1520725095, 8545396208, 48138689272, 271768537793, 1537212749339, 8709732836238, 49423440297324, 280835252080651, 1597731101233763

Offset: 0

Paul Barry, Oct 15 2005

Diagonal sums of A113139.

a(n)=sum{k=0..floor(n/2), sum{j=0..n-k, C(n-2k, j)C(n-k+j, k+j)}}.

cp's OEIS Frontend

A264772 Triangle T(n,k) = binomial(3*n - 2*k, 2*n - k), 0 <= k <= n.

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A190666 Number of walks from (0,0) to (n+3,n) which take steps from {E, N, NE}.

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A110171 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n that start with exactly k (0,1) steps (or, equivalently, with exactly k (1,0) steps).

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A264773 Triangle T(n,k) = binomial(4*n - 3*k, 3*n - 2*k), 0 <= k <= n.

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A264774 Triangle T(n,k) = binomial(5*n - 4*k, 4*n - 3*k), 0 <= k <= n.

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A113141 Inverse of a Delannoy related triangle.

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A113140 Diagonal sums of a Delannoy related triangle.

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A264772 Triangle T(n,k) = binomial(3n - 2k, 2*n - k), 0 <= k <= n.

A264773 Triangle T(n,k) = binomial(4n - 3k, 3n - 2k), 0 <= k <= n.

A264774 Triangle T(n,k) = binomial(5n - 4k, 4n - 3k), 0 <= k <= n.