A008288 -id:A008288 - cp's OEIS frontend

A216182 Riordan array ((1+x)/(1-x)^2, x(1+x)^2/(1-x)^2).

1, 3, 1, 5, 7, 1, 7, 25, 11, 1, 9, 63, 61, 15, 1, 11, 129, 231, 113, 19, 1, 13, 231, 681, 575, 181, 23, 1, 15, 377, 1683, 2241, 1159, 265, 27, 1, 17, 575, 3653, 7183, 5641, 2047, 365, 31, 1, 19, 833, 7183, 19825, 22363, 11969, 3303, 481, 35, 1

Offset: 0

Philippe Deléham, Mar 11 2013

Triangle formed of odd-numbered columns of the Delannoy triangle A008288.

			Triangle begins
   1;
   3,   1;
   5,   7,    1;
   7,  25,   11,    1;
   9,  63,   61,   15,    1;
  11, 129,  231,  113,   19,    1;
  13, 231,  681,  575,  181,   23,   1;
  15, 377, 1683, 2241, 1159,  265,  27,  1;
  17, 575, 3653, 7183, 5641, 2047, 365, 31, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. (columns:) A005408, A001845, A001847, A001849, A008419.
Cf. Diagonals: A000012, A004767, A060820.
Cf. A008288 (Delannoy triangle), A114123 (even-numbered columns of A008288).

A216182[n_, k_]:= Hypergeometric2F1[-n +k, -2*k-1, 1, 2];
Table[A216182[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 19 2021 *)

def A216182(n,k): return simplify( hypergeometric([-n+k, -2*k-1], [1], 2) )
flatten([[A216182(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 19 2021

T(2n, n) = A108448(n+1).
Sum_{k=0..n} T(n,k) = A073717(n+1).
From G. C. Greubel, Nov 19 2021: (Start)
T(n, k) = A008288(n+k+1, 2*k+1).
T(n, k) = hypergeometric([-n+k, -2*k-1], [1], 2). (End)

A346539 a(n) is the number of paths in the Z X Z grid joining (0,0) and (n,n) each of whose steps increases the Euclidean distance to the origin and has coordinates with absolute value at most 1.

Original entry on oeis.org

1, 3, 25, 241, 2545, 28203, 322681, 3776275, 44947503, 542097295, 6607714859, 81247609095, 1006335719467, 12542292874825, 157159924565801, 1978517963096763, 25010881408459855, 317327992746937599, 4039340709637022007, 51569571332132589961, 660140626022179390983

Offset: 0

José María Grau Ribas, Sep 10 2021

nonn

All terms are odd.

Alois P. Heinz, Table of n, a(n) for n = 0..886
Alois P. Heinz, Animation of a(3) = 241 paths

Main diagonal of A346538.
Column k=2 of A347811.
Cf. A008288, A001850, A225042.

b:= proc(n, k) option remember; `if`([n, k]=[0$2], 1, add(add(
     `if`(i^2+j^2 b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 12 2021

rodean[{m_, n_}] := Select[ Complement[ Flatten[Table[{m, n} + {s, t}, {s, -1, 1}, {t, -1, 1}], 1] // Union, {{m, n}}], #[[1]]^2 + #[[2]]^2 < m^2 + n^2 &];
$RecursionLimit=10^6; Clear[T]; T[{0, 0}]=1; T[{m_,n_}]:= T[{m,n}]= Sum[T[rodean[{m,n}][[i]]],{i,Length[rodean[{m, n}]]}]; Table[T[{n,n}],{n, 0,22}]
(* Second program: *)
b[n_, k_] := b[n, k] = If[{n, k} == {0, 0}, 1, Sum[Sum[If[i^2 + j^2 < n^2 + k^2, b@@Sort[{i, j}], 0], {j, k-1, k+1}], {i, n-1, n+1}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Nov 03 2021, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = 1/6*(19009+153*sqrt(17))^(1/3) + 356/(3*(19009+153*sqrt(17))^(1/3)) + 14/3 = 13.56165398271839628518... and c = 2.3842296614800994817864695565477260682981556338086519... . - Vaclav Kotesovec, Sep 13 2021

A053805 Expansion of (1 + x)^12 / (1 - x)^13.

Original entry on oeis.org

1, 25, 313, 2625, 16641, 85305, 369305, 1392065, 4673345, 14218905, 39753273, 103274625, 251595969, 579168825, 1267854873, 2653649025, 5334940545, 10343052825, 19403906105, 35330137025, 62596382081, 108167252025, 182668423833, 302016962625, 489658242241

Offset: 0

R. K. Guy, Apr 07 2000

This is row/column 12 of the Delannoy numbers array, A008288, which is the main entry for these numbers, listing many more properties. - Peter Munn, Jan 05 2023

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A008288, A240876.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)^12/(1-x)^13)); // Bruno Berselli, Apr 17 2014

CoefficientList[Series[(1 + x)^12/(1 - x)^13, {x, 0, 30}], x] (* Bruno Berselli, Apr 17 2014 *)

Vec((1+x)^12/(1-x)^13+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

G.f.: (1+x)^12/(1-x)^13.
a(n) = A240876(n) + 2*Sum_{i=0..n-1} A240876(i) for n>0, a(0)=1. - Bruno Berselli, Apr 17 2014
a(n) = 13*a(n-1) - 78*a(n-2) + 286*a(n-3) - 715*a(n-4) + 1287*a(n-5) - 1716*a(n-6) + 1716*a(n-7) - 1287*a(n-8) + 715*a(n-9) - 286*a(n-10) + 78*a(n-11) - 13*a(n-12) + a(n-13). - Wesley Ivan Hurt, Jul 09 2025

A064643 Bidirectional 'Delannoy' variation of the Boustrophedon transform applied to all 1's sequence: Fill an triangular array in alternating directions. Let the first element of each row in that direction be equal to 1. Each next entry is given by T(n,k) = T(n,k +/- 1) + T(n-1,k-1) + T(n-1,k) + T(n-2,k-1), where the +/- depends on which is the previous element in the direction one is filling in the row. The final number of row n gives a(n).

Original entry on oeis.org

1, 2, 6, 22, 105, 631, 4603, 39469, 388870, 4327322, 53670985, 734069672, 10975379510, 178080287645, 3116286236549, 58502460526469

Offset: 0

Floor van Lamoen, Oct 03 2001

nonn

Index entries for sequences related to boustrophedon transform

Cf. A064641. Table: A064644, Delannoy numbers A008288.

A101167 Nontrivial Delannoy numbers that are primes.

Original entry on oeis.org

13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, 1201, 1289, 1301, 1741, 1861, 2113, 2381, 2521, 3121, 3613, 4513, 5101, 5641, 7321, 8581, 9661, 9941, 10513, 11969, 12641, 13613, 14281, 14621, 15313, 16381, 19013, 19801, 20201, 21013, 21841

Offset: 1

Reinhard Zumkeller, Dec 03 2004

nonn

Let D and T be defined as in A008288: then D(n,1)=D(1,n)=T(n,1)=T(n,n-1)=2*n+1, therefore all odd primes are Delannoy numbers; the sequence contains only primes of form D(n,k) with n>1 and k>1, resp. T(n,k) with 2<=k<=n-2.

apart from the first term A092830 is a subsequence.

			D(2,7)=T(9,2)=113=A000040(30), therefore 113 is a term.

Eric Weisstein's World of Mathematics, Delannoy Number

A112475 Riordan array (1/(1+x),x(1+x)/(1-x)).

Original entry on oeis.org

1, -1, 1, 1, 1, 1, -1, 1, 3, 1, 1, 1, 5, 5, 1, -1, 1, 7, 13, 7, 1, 1, 1, 9, 25, 25, 9, 1, -1, 1, 11, 41, 63, 41, 11, 1, 1, 1, 13, 61, 129, 129, 61, 13, 1, -1, 1, 15, 85, 231, 321, 231, 85, 15, 1, 1, 1, 17, 113, 377, 681, 681, 377, 113, 17, 1

Offset: 0

Paul Barry, Sep 07 2005

Equivalent to Delannoy triangle A008288 with prepended column 1,-1,1,-1,... Row sums are A111954. Diagonal sums are A112476. Inverse is A112477.

			Triangle starts:
   1;
  -1, 1;
   1, 1, 1;
  -1, 1, 3,  1;
   1, 1, 5,  5,  1;
  -1, 1, 7, 13,  7, 1;
   1, 1, 9, 25, 25, 9, 1;
  ...

Cf. A008288, A111954, A112476, A112477.

T[n_,k_]:=SeriesCoefficient[(x(1+x)/(1-x))^k/(1+x),{x,0,n}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, May 26 2024 *)

T(n,k) = Sum{j=0..n-k} C(k-1,j)*C(n-j-1,n-k-j).

A142977 Table of coefficients in the expansion of the rational function 1/{(1-x)^2 - y*(1+x)^2}.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 10, 19, 4, 1, 14, 51, 44, 5, 1, 18, 99, 180, 85, 6, 1, 22, 163, 476, 501, 146, 7, 1, 26, 243, 996, 1765, 1182, 231, 8, 1, 30, 339, 1804, 4645, 5418, 2471, 344, 9, 1, 34, 451, 2964, 10165, 17718, 14407, 4712, 489, 10

Offset: 0

Peter Bala, Jul 15 2008

The row entries are the figurate numbers of the odd dimensional cross polytopes. See A142978 for the complete table of figurate numbers of n-dimensional cross polytopes. The rows are the partial sums of the even-numbered rows of the square array of Delannoy numbers A008288.

			The square array begins
 n\k| 0...1....2.....3.....4.......5
------------------------------------
 .0.| 1...2....3.....4......5......6 ... A000027
 .1.| 1...6...19....44.....85....146 ... A005900
 .2.| 1..10...51...180....501...1182 ... A069038
 .3.| 1..14...99...476...1765...5418 ... A099193
 .4.| 1..18..163...996...4645..17718 ... A099196
 .5.| 1..22..243..1804..10165..46530 ... A300624
 ...

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.

Cf. A005900 (row 1), A008288, A069038 (row 2), A099193 (row 3), A099196 (row 4), A300624 (row 5), A142978, A142983.

with(combinat): T:=(n,k) -> add(binomial(2n,k-j)*binomial(2n+j+1,j), j = 0..k): for n from 0 to 9 do seq(T(n,k), k = 0..9) end do;

T(n,k) = Sum_{j = 0..k} C(2*n, k-j)*C(2*n+j+1, j).
O.g.f.: 1/{(1 - x)^2 - y*(1 + x)^2} = Sum_{n, k >= 0} T(n,k)*x^k*y^n = 1/(1 - y) * Sum_{m >= 0} U(m, (1 + y)/(1 - y))*x^m, where U(m, y) denotes the m-th Chebyshev polynomial of the second kind.
O.g.f. row n: (1 + x)^(2*n)/(1 - x)^(2*n+2).
O.g.f. column k: 1/(1 - y)*U(k, (1 + y)/(1 - y)).
The entries in the n-th row appear in the series acceleration formula for the constant log(2): Sum_{k >= 1} (-1)^(k+1)/(T(n,k)*T(n,k+1)) = 1 + (4*n + 2)*( log(2) - (1 - 1/2 + 1/3 - ... + 1/(2*n + 1)) ).
For example, n = 1 gives log(2) = 4/6 + (1/6)*( 1/(1*6) - 1/(6*19) + 1/(19*44) - 1/(44*85) + ... ). See A142983 for further details.

Restored missing program. - Peter Bala, Oct 02 2008

A225413 Triangle read by rows: T(n,k) = (A101164(n,k) - A014473(n,k))/2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 6, 12, 6, 0, 0, 0, 0, 10, 30, 30, 10, 0, 0, 0, 0, 15, 60, 91, 60, 15, 0, 0, 0, 0, 21, 105, 215, 215, 105, 21, 0, 0, 0, 0, 28, 168, 435, 590, 435, 168, 28, 0, 0, 0, 0, 36, 252, 791, 1365, 1365, 791, 252, 36, 0, 0

Offset: 0

Jeremy Gardiner, Jul 28 2013

Has opposite parity to A140356, A155454.

			Triangle begins as:
  0;
  0,  0;
  0,  0,  0;
  0,  0,  0,   0;
  0,  0,  1,   0,    0;
  0,  0,  3,   3,    0,    0;
  0,  0,  6,  12,    6,    0,    0;
  0,  0, 10,  30,   30,   10,    0,    0;
  0,  0, 15,  60,   91,   60,   15,    0,    0;
  0,  0, 21, 105,  215,  215,  105,   21,    0,    0;
  0,  0, 28, 168,  435,  590,  435,  168,   28,    0,   0;
  0,  0, 36, 252,  791, 1365, 1365,  791,  252,   36,   0,  0;
  0,  0, 45, 360, 1330, 2800, 3571, 2800, 1330,  360,  45,  0,  0;
  0,  0, 55, 495, 2106, 5250, 8197, 8197, 5250, 2106, 495, 55,  0,  0;

Reinhard Zumkeller, Rows n = 0..100 of table, flattened

Cf. A000129, A007318, A008288, A014473, A101164, A121262, A140356, A155454.
3rd column = A000217 (triangular numbers).
4th column = A027480 (n(n+1)(n+2)/2).

a225413 n k = a225413_tabl !! n !! k
a225413_row n = a225413_tabl !! n
a225413_tabl = map (map (`div` 2)) $
               zipWith (zipWith (-)) a101164_tabl a014473_tabl
-- Reinhard Zumkeller, Jul 30 2013

A008288:= func< n,k | (&+[Binomial(n-j, j)*Binomial(n-2*j, k-j): j in [0..k]]) >;
A225413:= func< n,k | (A008288(n,k) - 2*Binomial(n,k) + 1)/2 >;
[A225413(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 08 2024

T[n_, k_]:= ((-1)^(n-k)*Hypergeometric2F1[-n+k,k+1,1,2] - 2*Binomial[n, k] +1)/2;
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 08 2024 *)

def A008288(n,k): return sum(binomial(n-j,j)*binomial(n-2*j,k-j) for j in range(k+1))
def A225413(n,k): return (A008288(n,k) -2*binomial(n,k) +1)//2
flatten([[A225413(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 08 2024

T(n, k) = (A101164(n,k) - A014473(n,k))/2.
T(n, k) = (A008288(n,k) - 2*A007318(n,k) + 1)/2.
From G. C. Greubel, Apr 08 2024: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = (A000129(n+1) + n + 1 - 2^(n+1))/2.
Sum_{k=0..n} (-1)^k*T(n, k) = A121262(n) - [n=0]. (End)

A227964 Triangle where the g.f. of row n equals (1-x-x^2+x^3)^n and terms T(n,k) are read by rows n>=0, k=0..3*n.

Original entry on oeis.org

1, 1, -1, -1, 1, 1, -2, -1, 4, -1, -2, 1, 1, -3, 0, 8, -6, -6, 8, 0, -3, 1, 1, -4, 2, 12, -17, -8, 28, -8, -17, 12, 2, -4, 1, 1, -5, 5, 15, -35, -1, 65, -45, -45, 65, -1, -35, 15, 5, -5, 1, 1, -6, 9, 16, -60, 24, 116, -144, -66, 220, -66, -144, 116, 24, -60, 16, 9, -6, 1, 1, -7, 14, 14, -91, 77, 168, -344, -14, 546, -364, -364, 546, -14, -344, 168, 77, -91, 14, 14, -7, 1

Offset: 0

Paul D. Hanna, Aug 01 2013

			Triangle begins:
1;
1, -1, -1, 1;
1, -2, -1, 4, -1, -2, 1;
1, -3, 0, 8, -6, -6, 8, 0, -3, 1;
1, -4, 2, 12, -17, -8, 28, -8, -17, 12, 2, -4, 1;
1, -5, 5, 15, -35, -1, 65, -45, -45, 65, -1, -35, 15, 5, -5, 1;
1, -6, 9, 16, -60, 24, 116, -144, -66, 220, -66, -144, 116, 24, -60, 16, 9, -6, 1;
1, -7, 14, 14, -91, 77, 168, -344, -14, 546, -364, -364, 546, -14, -344, 168, 77, -91, 14, 14, -7, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..2500

Cf. A192205.

{T(n,k)=polcoeff((1-x-x^2+x^3 +x*O(x^k))^n,k)}
for(n=0,10,for(k=0,3*n,print1(T(n,k),", "));print(""))

Sum_{k=0..3*n} |T(n,k)| = A192205(n).
Sum_{k=0..3*n} T(n,k)^2 = binomial(4*n,n).
Sum_{k=0..3*n} T(n,k) * binomial(3*n,k) = (-1)^n * binomial(4*n,n).
Sum_{k=0..3*n} T(n,k) * binomial(2*n+k,k) = 2^n.
Sum_{k=0..3*n} T(n,k) * binomial(3*n+k,k) = A008288(3*n,n), where A008288 is the Delannoy array (see A026001).

A229995 Array of coefficients of numerator polynomials of the rational function p(n, x + 1/x), where p(n,x) is the Fibonacci polynomial defined by p(1,x) = 1, p(2,x) = x, p(n,x) = x*p(n-1,x) + p(n-2,x).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 5, 0, 5, 0, 1, 1, 0, 7, 0, 13, 0, 7, 0, 1, 1, 0, 9, 0, 25, 0, 25, 0, 9, 0, 1, 1, 0, 11, 0, 41, 0, 63, 0, 41, 0, 11, 0, 1, 1, 0, 13, 0, 61, 0, 129, 0, 129, 0, 61, 0, 13, 0, 1, 1, 0, 15, 0, 85, 0, 231, 0, 321, 0, 231, 0, 85, 0

Offset: 0

Clark Kimberling, Nov 07 2013

Deleting the 0's leaves A008288 (Delannoy numbers as a triangle). Define q(n,x) = p(n, x + 1/x). If r is a zero of p(n,x) then (1/2)*(r +- sqrt(r^2 - 4)) are zeros of q(n,x).

			First 4 rows:
  1
  1 0 1
  1 0 3 0 1
  1 0 5 0 5 0 1

Cf. A230002, A008288.

p[n_, x_] := p[x] = Fibonacci[n, x]; Table[p[n, x], {n, 1, 10}]
f[n_, x_] := f[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x + 1/x]]]
g[n_, x_] := g[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x - 1/x]]]
h[n_, x_] := h[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x + 1 + 1/x]]]
t1 = Flatten[Table[CoefficientList[f[n, x], x], {n, 1, 12}]];  (* A229995 *)
t2 = Flatten[Table[CoefficientList[g[n, x], x], {n, 1, 12}]];  (* A230002 *)
t3 = Flatten[Table[CoefficientList[h[n, x], x], {n, 1, 12}]];  (* A059317 *)

cp's OEIS Frontend

A216182 Riordan array ((1+x)/(1-x)^2, x(1+x)^2/(1-x)^2).

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A346539 a(n) is the number of paths in the Z X Z grid joining (0,0) and (n,n) each of whose steps increases the Euclidean distance to the origin and has coordinates with absolute value at most 1.

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A053805 Expansion of (1 + x)^12 / (1 - x)^13.

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A101167 Nontrivial Delannoy numbers that are primes.

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A112475 Riordan array (1/(1+x),x(1+x)/(1-x)).

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A142977 Table of coefficients in the expansion of the rational function 1/{(1-x)^2 - y*(1+x)^2}.

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A225413 Triangle read by rows: T(n,k) = (A101164(n,k) - A014473(n,k))/2.

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