A026000 -id:A026000 - cp's OEIS frontend

A026001 a(n) = T(3n,n), where T = Delannoy triangle (A008288).

1, 7, 85, 1159, 16641, 246047, 3707509, 56610575, 872893441, 13560999991, 211939849045, 3328419072535, 52481589415425, 830317511708367, 13174519143904245, 209559710593266719, 3340604559333629953, 53354776911196959335, 853607938952248383829, 13677336690921351929767

Offset: 0

Clark Kimberling

If the Delannoy triangle is defined by the Maple code in A008288, this is A008288(n, 3*n-2), n >= 1. - N. J. A. Sloane, Oct 29 2006

Seiichi Manyama, Table of n, a(n) for n = 0..823
Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).

Cf. A008288, A001850, A026000, A331329.

F := (2-t)/(3*t^2-8*t+2);  G := t*(t-1)^3/(t-2);  Ginv := RootOf(numer(G-x),t); ogf := series(eval(F, t=Ginv), x=0, 25); # Mark van Hoeij, Oct 30 2011

a[n_] := Binomial[4 n, n] Hypergeometric2F1[-3 n, -n, -4 n, -1];
Array[a, 20, 0] (* Peter Luschny, Jan 31 2020 *)

G.f.: F(G^(-1)(x)) where F = (2-t)/(3*t^2-8*t+2) and G = t*(t-1)^3/(t-2). - Mark van Hoeij, Oct 30 2011
From Peter Bala, Jan 29 2020: (Start)
a(n) = Sum_{k = 0..n}  C(n,k)*C(3*n+k,n).
a(n) = Sum_{k = 0..n}  C(n,k)*C(4*n-k,n).
a(n) = Sum_{k = 0..n}  C(3*n,n-k)*C(3*n+k,k).
a(n) = Sum_{k = 0..n} 2^k*C(n,k)*C(3*n,k).
a(n) = Sum_{k = 0..n} C(4*n-k,k)*C(4*n-2*k,n-k).
3*n*(3*n - 1)*(3*n - 2)*(70*n^2 - 189*n + 127)*a(n) = 2*(15610*n^5 - 65562*n^4 + 102255*n^3 - 72864*n^2 + 23369*n - 2640)*a(n-1) - 3*(n - 1)* (3*n - 4)*(3*n - 5)*(70*n^2 - 49*n + 8)*a(n-2) with a(0) = 1, a(1) = 7.
(End)
a(n) = binomial(4*n, n)*hypergeom([-3*n, -n], [-4*n], -1). - Peter Luschny, Jan 31 2020
a(n) ~ sqrt(1 + 13/(4*sqrt(10))) * (223 + 70*sqrt(10))^n / (sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Feb 13 2021
D-finite with recurrence +435*n*(3*n-1)*(3*n-2)*a(n) +(-53978*n^3+43545*n^2+39923*n-35580)*a(n-1) +3*(-57648*n^3+321915*n^2-580787*n+339980)*a(n-2) +9*(1634*n^3-11365*n^2+27137*n-22546)*a(n-3) -27*(3*n-10)*(3*n-11)*(n-3)*a(n-4)=0. - R. J. Mathar, Aug 01 2022

Corrected and extended by N. J. A. Sloane, Oct 29 2006

A339710 a(n) = Sum_{k=0..n} binomial(n, k)binomial(2n + k, k)*2^k.

Original entry on oeis.org

1, 7, 81, 1051, 14353, 201807, 2891409, 41976627, 615371169, 9089130967, 135048608401, 2016306678987, 30224723308081, 454603719479839, 6857319231939537, 103694587800440931, 1571449259865571137, 23860205774602899111, 362897293035114695121, 5527773456878667951483

Offset: 0

Yifan Zhang, Dec 13 2020

nonn

Frits Beukers, Some Congruences for Apery Numbers, Mathematisch Instituut, University of Leiden, 1983, pages 1-2.

Seiichi Manyama, Table of n, a(n) for n = 0..838
F. Beukers, Some congruences for the Apery numbers, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155. local copy

Cf. A000079 (Sum(binomial(n, k))), A000984 (Sum(binomial(n, k)^2)), A026375 (Sum(binomial(n, k)*binomial(2*k, k))), A001850 (Sum(binomial(n, k)*binomial(n+k, k))), A005809 (Sum(binomial(n, k)*binomial(2*n, k))).

Cf. A026000, A114496.

Table[Sum[Binomial[n,k]*Binomial[2n+k,k]*2^k,{k,0,n}],{n,0,20}] (* or *)
Table[Hypergeometric2F1[-n,1+2 n,1,-2],{n,0,20}] (* Stefano Spezia, Dec 17 2020 *)

a(n) = sum(k=0, n, binomial(n, k)*binomial(2*n + k, k)*2^k); \\ Michel Marcus, Feb 12 2021

a(n) = 2F1([-n, 1 + 2*n], [1], -2), where 2F1 is the hypergeometric function. - Stefano Spezia, Dec 17 2020
From Vaclav Kotesovec, May 11 2021: (Start)
Recurrence: 3*n*(2*n - 1)*(26*n - 35)*a(n) = (2444*n^3 - 5734*n^2 + 3830*n - 729)*a(n-1) - (n-1)*(2*n - 3)*(26*n - 9)*a(n-2).
a(n) ~ sqrt(3/8 + 11/(8*sqrt(13))) * ((47 + 13*sqrt(13))/6)^n / sqrt(Pi*n). (End)

More terms from Stefano Spezia, Dec 17 2020

A331329 a(n) = binomial(5n, n)hypergeom([-4n, -n], [-5n], -1).

Original entry on oeis.org

1, 9, 145, 2625, 50049, 982729, 19665841, 398796225, 8166636545, 168502295625, 3497529199185, 72949645000065, 1527671538372225, 32100078290806665, 676451066002195825, 14290577765009652865, 302557549412667613185, 6417968867896642617225, 136371773642235542394385

Offset: 0

Peter Luschny, Jan 31 2020

nonn

Special case of generalized Delannoy numbers (see cross-references):

T(n, k) = binomial(k*n, n)*hypergeom([(1-k)*n, -n], [-k*n], -1).

Seiichi Manyama, Table of n, a(n) for n = 0..747
Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).

Cf. A001850 (k=2), A026000 (k=3), A026001 (k=4), this sequence (k=5), A341491 (k=6).

Cf. A008288, A181675, A341476, A341477.

a[n_] := Binomial[5 n, n] Hypergeometric2F1[-4 n, -n, -5 n, -1];
Array[a, 19, 0]

a(n) ~ sqrt(5 + 21/sqrt(17)) * (349 + 85*sqrt(17))^n / (sqrt(Pi*n) * 2^(5*n + 2)). - Vaclav Kotesovec, Feb 13 2021

A341491 a(n) = binomial(6n, n) hypergeom([-5n, -n], [-6n], -1).

Original entry on oeis.org

1, 11, 221, 4991, 118721, 2908411, 72616013, 1837271615, 46943003137, 1208483403179, 31297149356221, 814471993937855, 21281058718583873, 557930580979801755, 14669716953106628781, 386675596518995000191, 10214494658006725840897, 270345191656309313382475

Offset: 0

Vaclav Kotesovec, Feb 13 2021

nonn

In general, for k > 1, binomial(k*n, n) * hypergeom([(1-k)*n, -n], [-k*n], -1) ~ sqrt((k + (k^2 - k + 1) / sqrt(k^2 - 2*k + 2)) / (4*(k-1)*Pi*n)) * ((A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k-1)^(k-1))^n.

Michael De Vlieger, Table of n, a(n) for n = 0..697
Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).

Cf. A001850, A026000, A026001, A331329, A341476, A341477.

Table[Binomial[6*n, n] * Hypergeometric2F1[-5*n, -n, -6*n, -1], {n,0,20}]

a(n) ~ sqrt((6 + 31/sqrt(26))/(20*Pi*n)) * (42671 + 8346*sqrt(26))^n / 5^(5*n).

A341476 Coefficients related to the asymptotics of generalized Delannoy numbers.

Original entry on oeis.org

1, 3, 22, 223, 2792, 42671, 761984, 15707707, 365122688, 9491746747, 271962399232, 8539383210711, 290937486190592, 10710312199270503, 422984587596455936, 17864076455770831219, 802450164859200372736, 38242916911507537149427, 1925477163696152909447168, 102213291475268656299164879

Offset: 1

Vaclav Kotesovec, Feb 13 2021

nonn

			Lim_{n->infinity} A001850(n)^(1/n) = (    3 +    2 * sqrt(1^2 + 1)) / 1^1.
Lim_{n->infinity} A026000(n)^(1/n) = (   22 +   10 * sqrt(2^2 + 1)) / 2^2.
Lim_{n->infinity} A026001(n)^(1/n) = (  223 +   70 * sqrt(3^2 + 1)) / 3^3.
Lim_{n->infinity} A331329(n)^(1/n) = ( 2792 +  680 * sqrt(4^2 + 1)) / 4^4.
Lim_{n->infinity} A341491(n)^(1/n) = (42671 + 8346 * sqrt(5^2 + 1)) / 5^5.

Vaclav Kotesovec, Table of n, a(n) for n = 1..100

Cf. A008288, A001850, A026000, A026001, A331329, A341491, A341477.

Lim_{n->infinity} (binomial(k*n, n) * hypergeom([(1-k)*n, -n], [-k*n], -1))^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k-1)^(k-1), for k>1.
Lim_{n->infinity} hypergeom([(1-k)*n, -n], [-k*n], -1)^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / k^k.
For k > 1, A341476(k)^2 - ((k-1)^2 + 1) * A341477(k)^2 = (-1)^k * (k-1)^(2*k-2).
Lim_{k->infinity} (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k * (k-1)^(k-1)) = 2*exp(1).
a(n) ~ n^n.

A341477 Coefficients related to the asymptotics of generalized Delannoy numbers.

Original entry on oeis.org

0, 2, 10, 70, 680, 8346, 125504, 2218350, 45335680, 1047314578, 27079557632, 772687787510, 24172386314240, 821114930966890, 30146801401143296, 1187943632192716894, 50068690149298438144, 2245175953053786221730, 106828553482726336102400, 5371204894269759411503910

Offset: 1

Vaclav Kotesovec, Feb 13 2021

nonn

			Lim_{n->infinity} A001850(n)^(1/n) = (    3 +    2 * sqrt(1^2 + 1)) / 1^1.
Lim_{n->infinity} A026000(n)^(1/n) = (   22 +   10 * sqrt(2^2 + 1)) / 2^2.
Lim_{n->infinity} A026001(n)^(1/n) = (  223 +   70 * sqrt(3^2 + 1)) / 3^3.
Lim_{n->infinity} A331329(n)^(1/n) = ( 2792 +  680 * sqrt(4^2 + 1)) / 4^4.
Lim_{n->infinity} A341491(n)^(1/n) = (42671 + 8346 * sqrt(5^2 + 1)) / 5^5.

Vaclav Kotesovec, Table of n, a(n) for n = 1..100

Cf. A008288, A001850, A026000, A026001, A331329, A341491, A341476.

Lim_{n->infinity} (binomial(k*n, n) * hypergeom([(1-k)*n, -n], [-k*n], -1))^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k-1)^(k-1), for k>1.
Lim_{n->infinity} hypergeom([(1-k)*n, -n], [-k*n], -1)^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / k^k.
For k > 1, A341476(k)^2 - ((k-1)^2 + 1) * A341477(k)^2 = (-1)^k * (k-1)^(2*k-2).
Lim_{k->infinity} (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k * (k-1)^(k-1)) = 2*exp(1).
a(n) ~ n^(n-1).

A099605 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle A034870 of coefficients in successive powers of the trinomial (1+2*x+x^2), omitting leading zeros.

Original entry on oeis.org

1, 2, 2, 1, 5, 4, 4, 16, 20, 8, 1, 14, 41, 44, 16, 6, 50, 146, 198, 128, 32, 1, 27, 155, 377, 456, 272, 64, 8, 112, 560, 1408, 1992, 1616, 704, 128, 1, 44, 406, 1652, 3649, 4712, 3568, 1472, 256, 10, 210, 1572, 6084, 14002, 20330, 18880, 10912, 3584, 512, 1, 65

Offset: 0

Paul D. Hanna, Oct 25 2004

Row sums form A099606, where A099606(n) = Pell(n+1)*2^[(n+1)/2]. Central coefficients of even-indexed rows form A026000, where A026000(n) = T(2n,n), where T = Delannoy triangle (A008288). Antidiagonal sums form A099607.

			Rows begin:
[1],
[2,2],
[1,5,4],
[4,16,20,8],
[1,14,41,44,16],
[6,50,146,198,128,32],
[1,27,155,377,456,272,64],
[8,112,560,1408,1992,1616,704,128],
[1,44,406,1652,3649,4712,3568,1472,256],
[10,210,1572,6084,14002,20330,18880,10912,3584,512],
[1,65,870,5202,17469,36365,48940,42800,23552,7424,1024],...
The binomial transform of row 2 equals column 2 of A034870:
BINOMIAL[1,5,4] = [1,6,15,28,45,66,91,120,153,...].
The binomial transform of row 3 equals column 3 of A034870:
BINOMIAL[4,16,20,8] = [4,20,56,120,220,364,560,...].
The binomial transform of row 4 equals column 4 of A034870:
BINOMIAL[1,14,41,44,16] = [1,15,70,210,495,1001,...].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A099602, A099605, A034870, A000129.

CoefficientList[CoefficientList[Series[(1 + 2*(y + 1)*x - (y + 1)*x^2)/(1 - (2*y + 1)*(2*y + 2)*x^2 + (y + 1)^2*x^4), {x, 0, 49}, {y, 0, 49}], x],
  y] // Flatten (* G. C. Greubel, Apr 14 2017 *)

{T(n,k)=polcoeff(polcoeff((1+2*(y+1)*x-(y+1)*x^2)/(1-(2*y+1)*(2*y+2)*x^2+(y+1)^2*x^4)+x*O(x^n),n,x)+y*O(y^k),k,y)}

G.f.: (1+2*(y+1)*x-(y+1)*x^2)/(1-(2*y+1)*(2*y+2)*x^2+(y+1)^2*x^4). T(n, n) = 2^n.

A114164 Riordan array (1/(1-2x), x(1-x)/(1-2x)^2).

Original entry on oeis.org

1, 2, 1, 4, 5, 1, 8, 18, 8, 1, 16, 56, 41, 11, 1, 32, 160, 170, 73, 14, 1, 64, 432, 620, 377, 114, 17, 1, 128, 1120, 2072, 1666, 704, 164, 20, 1, 256, 2816, 6496, 6608, 3649, 1178, 223, 23, 1, 512, 6912, 19392, 24192, 16722, 7001, 1826, 291, 26, 1, 1024, 16640, 55680, 83232, 69876, 36365, 12235, 2675, 368, 29, 1

Offset: 0

Paul Barry, Nov 15 2005

Row sums are A081567. Diagonal sums are A085810. Product of Pascal triangle A007318 and Morgan-Voyce triangle A085478.

Unsigned version of A123876. - Philippe Deléham, Oct 25 2007

			Triangle begins:
   1;
   2,   1;
   4,   5,   1;
   8,  18,   8,  1;
  16,  56,  41, 11,  1;
  32, 160, 170, 73, 14, 1;
  ...

Cf. A081567, A085810, A007318, A085478, A123876.

T(2n,n) gives A026000.

Number triangle T(n,k) = Sum_{j=0..n} C(n, j)*C(j+k, 2k);
T(n,k) = Sum_{j=0..n} C(n, k+j)*C(k, k-j)*2^(n-k-j);
T(n,k) = Sum_{j=0..n-k} C(n+k-j, n-k-j)*C(k, j)*(-1)^j*2^(n-k-j).
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 4*T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 2, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 17 2014

More terms from Michel Marcus, Sep 09 2024

A181675 V(n,n^2), where V is the number of integer points in an n-dimensional sphere of Lee-radius n^2 centered at the origin.

Original entry on oeis.org

3, 41, 1159, 50049, 2908411, 212358985, 18665359119, 1917971421185, 225555471838387, 29871434052884841, 4398867465890529303, 712959801840558794625, 126115813138335816685995

Offset: 2

Antonio Campello, Nov 04 2010

Since V(n,d) is symmetric, we have V(n,n^2) = V(n^2,n).

Solomon W. Golomb and Lloyd R. Welch, Perfect Codes in the Lee Metric and the Packing of Polyominoes, SIAM Journal on Applied Mathematics Vol. 18, No. 2 (Mar. 1970), pp. 302-317.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

V(n, n) = A001850, V(n, 2n) = A026000 and V(n, 3n) = A026001.

Array[Sum[2^j*Binomial[#1, j] Binomial[#2, j], {j, 0, Min[#1, #2]}] & @@ {#, #^2} &, 13] (* Michael De Vlieger, Jul 05 2019 *)

V(n,d) = Sum_{j=0..min(n,d)} 2^j * binomial(n,j)*binomial(d,j).

a(n) ~ exp(n-2) * (2*n)^(n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Feb 13 2021

A357613 Triangle read by rows T(n, k) = binomial(2n, k) binomial(3n - k, 2n).

Original entry on oeis.org

1, 3, 2, 15, 20, 6, 84, 168, 105, 20, 495, 1320, 1260, 504, 70, 3003, 10010, 12870, 7920, 2310, 252, 18564, 74256, 120120, 100100, 45045, 10296, 924, 116280, 542640, 1058148, 1113840, 680680, 240240, 45045, 3432

Offset: 0

F. Chapoton, Oct 06 2022

Each line should be the f-vector of a cellular complex. The sequence seems to give the coefficients in a binomial basis of the integer-valued polynomials (x+1)*(x+2)*...*(x+2*n)*(x+1)*(x+2)*...*(x+n)/(n!*(2n)!).

The precise expansion is (x+1)*(x+2)*...*(x+2*n)*(x+1)*(x+2)*...*(x+n)/(n!*(2*n)!) = Sum_{k = 0..n} (-1)^k*T(n,k)*binomial(x+3*n-k, 3*n-k), as can be verified using the WZ algorithm. For example, n = 3 gives (x+1)^2*(x+2)^2*(x+3)^2*(x+4)*(x+5)*(x+6)/(3!*6!) = 84*binomial(x+9, 9) - 168*binomial(x+8, 8) + 105*binomial(x+7, 7) - 20*binomial(x+6, 6). - Peter Bala, Jun 25 2023

			As a triangle of numbers, this starts with
  1;
  3, 2;
  15, 20, 6;
  84, 168, 105, 20;
  495, 1320, 1260, 504, 70.
Here is an example for n=1 as coefficients (up to sign) in the binomial basis of integer-valued polynomials:
(x+1)*(x+2)*(x+1)/2 = 3*binomial(x+3,3)-2*binomial(x+2,2).

Row sums A026000. Cf. A000984, A005809 (k=0), A144485 (k=1), A033282, A110608, A243660.

A357613 := proc(n,k)
    binomial(2*n,k)*binomial(3*n-k,2*n) ;
end proc:
seq(seq(A357613(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Jul 06 2023

Table[Binomial[2n,k]Binomial[3n-k,2n],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Oct 11 2023 *)

def a(n):
    return [binomial(2 * n, k) * binomial(3 * n - k, 2 * n)
            for k in range(n + 1)]

T(n,k) = binomial(2*n, k) * binomial(3*n - k, 2*n) for 0 <= k <= n

cp's OEIS Frontend

A026001 a(n) = T(3n,n), where T = Delannoy triangle (A008288).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A339710 a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*n + k, k)*2^k.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A331329 a(n) = binomial(5*n, n)*hypergeom([-4*n, -n], [-5*n], -1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A341491 a(n) = binomial(6*n, n) * hypergeom([-5*n, -n], [-6*n], -1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A341476 Coefficients related to the asymptotics of generalized Delannoy numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A341477 Coefficients related to the asymptotics of generalized Delannoy numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A099605 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle A034870 of coefficients in successive powers of the trinomial (1+2*x+x^2), omitting leading zeros.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A114164 Riordan array (1/(1-2x), x(1-x)/(1-2x)^2).

Views

Author

Keywords

Comments

Examples

Crossrefs

A339710 a(n) = Sum_{k=0..n} binomial(n, k)binomial(2n + k, k)*2^k.

A331329 a(n) = binomial(5n, n)hypergeom([-4n, -n], [-5n], -1).

A341491 a(n) = binomial(6n, n) hypergeom([-5n, -n], [-6n], -1).

A357613 Triangle read by rows T(n, k) = binomial(2n, k) binomial(3n - k, 2n).