A183160 -id:A183160 - cp's OEIS frontend

A371770 a(n) = Sum_{k=0..floor(n/3)} binomial(3n-3k-1,n-3*k).

1, 2, 10, 57, 338, 2057, 12741, 79914, 505954, 3226638, 20696685, 133382658, 862978221, 5601919325, 36467212610, 237974911737, 1556281907586, 10196788555859, 66921360130374, 439860632463462, 2895002186799453, 19077000179746293, 125849150650146714

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Robert Israel, Table of n, a(n) for n = 0..1202

Cf. A005809, A165817, A183160.

Cf. A371758, A371771, A371772.

f:= proc(n) local k; add(binomial(3*n-3*k-1,n-3*k),k=0..n/3) end proc:
map(f, [$0..30]); # Robert Israel, Feb 28 2025

a(n) = sum(k=0, n\3, binomial(3*n-3*k-1, n-3*k));

a(n) = [x^n] 1/((1-x^3) * (1-x)^(2*n)).
a(n) = binomial(3*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1/3-n, 2/3-n, 1-n], 1). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 18*n*(2*n - 1)*(13*n - 22)*(37*n - 51)*a(n) = 3*(40885*n^4 - 165468*n^3 + 229373*n^2 - 125562*n + 22680)*a(n-1) - (40885*n^4 - 165468*n^3 + 229373*n^2 - 125562*n + 22680)*a(n-2) + 3*(3*n - 5)*(3*n - 4)*(13*n - 9)*(37*n - 14)*a(n-3).
a(n) ~ 3^(3*n + 5/2) / (13 * sqrt(Pi*n) * 2^(2*n+1)). (End)

A199033 Number of ways to place n non-attacking bishops on a 2 X 2n board.

Original entry on oeis.org

1, 4, 22, 128, 771, 4744, 29618, 186880, 1188679, 7608764, 48953224, 316283264, 2050706932, 13336273528, 86953633242, 568221290496, 3720529001823, 24403423540348, 160314652983158, 1054635453261568, 6946703172803003, 45809043607167328, 302395650703501688

Offset: 0

Vaclav Kotesovec, Nov 02 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
V. Kotesovec, Non-attacking chess pieces

Cf. A002465, A191236, A219197, A183160.

[(&+[Binomial(2*n-j+1,j)*Binomial(n+j+1,n-j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 19 2019

Table[Sum[Binomial[2n-j+1,j]*Binomial[n+j+1,n-j],{j,0,n}],{n,0,25}]

A199033(n):=sum(binomial(n+k+1, n-k)*binomial(2*n-k+1,k),k,0,n)$ makelist(A199033(n),n,0,22); /* Martin Ettl, Nov 15 2012 */

{a(n)=sum(k=0, n, binomial(n+k+1, n-k)*binomial(2*n-k+1, k))}

{a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(G^2/(1-2*x*G^2-3*x^2*G^4), n)} \\ Paul D. Hanna, Nov 14 2012
for(n=0,25,print1(a(n),", "))

[sum(binomial(2*n-j+1,j)*binomial(n+j+1,n-j) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Feb 19 2019

Recurrence: (112*n^4 + 968*n^3 + 3048*n^2 + 4136*n + 2040)*a(n+2) = (728*n^4 + 5914*n^3 + 17550*n^2 + 22510*n + 10530)*a(n+1) + (189*n^4 + 1539*n^3 + 4578*n^2 + 5886*n + 2760)*a(n). - Vaclav Kotesovec, Oct 30 2011
a(n) = Sum_{j=0..n} (binomial(2n-j+1,j)*binomial(n+j+1,n-j)).
a(n) ~ 3^(3n+4)/2^(2n+5)/sqrt(3*Pi*n).
Self-convolution of A219197. - Paul D. Hanna, Nov 14 2012
G.f.: A(x) = G(x)^2 / (1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 14 2012
a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(2*n+2)). - Ilya Gutkovskiy, Oct 25 2017

Offset changed to 0 and a(0)=1 added by Paul D. Hanna, Nov 14 2012

A225006 Number of n X n 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

1, 2, 9, 50, 295, 1792, 11088, 69498, 439791, 2803658, 17978389, 115837592, 749321716, 4863369656, 31655226108, 206549749930, 1350638103791, 8848643946550, 58069093513635, 381650672631330, 2511733593767295, 16550500379912640, 109176697072162080

Offset: 0

R. H. Hardin, Apr 23 2013

nonn

Diagonal of A225010.

Number of unimodal maps [1..n]->[1..n+1], see example. - Joerg Arndt, May 10 2013

			Some solutions for n=3
..0..1..1....0..1..0....0..0..1....0..0..0....0..0..0....0..0..0....0..0..0
..1..1..1....0..1..0....1..1..1....0..0..0....0..0..0....0..1..0....0..0..1
..1..1..1....0..1..1....1..1..1....0..0..1....0..1..0....1..1..1....0..1..1
From _Joerg Arndt_, May 10 2013: (Start)
The a(2) = 9 unimodal maps [1,2]->[1,2,3] are
01:  [ 1 1 ]
02:  [ 1 2 ]
03:  [ 1 3 ]
04:  [ 2 1 ]
05:  [ 2 2 ]
06:  [ 2 3 ]
07:  [ 3 1 ]
08:  [ 3 2 ]
09:  [ 3 3 ]
(End)

G. C. Greubel and R. H. Hardin, Table of n, a(n) for n = 0..1000 (terms 1..51 from R. H. Hardin)

Cf. A088536 (unimodal maps [1..n]->[1..n]).

Cf. A183160, A371798.

a[n_] := Sum[Binomial[2d+n-1, n-1], {d, 0, n}]; Array[a, 30] (* Jean-François Alcover, Feb 17 2016, after Max Alekseyev *)

{ a(n) = polcoeff( (1+x+O(x^(2*n+1)))^(-n-1)/(1-x), 2*n) }

From Vaclav Kotesovec, May 22 2013: (Start)
Empirical: 4*n*(2*n-1)*(5*n-7)*a(n) = 2*(145*n^3 - 343*n^2 + 235*n - 48)*a(n-1) - 3*(3*n-4)*(3*n-2)*(5*n-2)*a(n-2).
a(n) ~ 3^(3*n+3/2)/(5*2^(2*n+1)*sqrt(Pi*n)). (End)
a(n) = A261668(n)+1.
a(n) = Sum_{d=0..n} binomial(2d+n-1,n-1). Also, a(n) is the coefficient of x^(2n) in (1+x)^(-n-1)/(1-x). - Max Alekseyev, Sep 14 2015
It appears that a(n) = Sum_{k = 0..2*n} (-1)^k*binomial(n+k,k). - Peter Bala, Oct 08 2021
From Seiichi Manyama, Apr 06 2024: (Start)
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-2*k-1,n-2*k).
a(n) = [x^n] 1/((1+x^2) * (1-x)^(2*n)). (End)

a(0)=1 prepended by Alois P. Heinz, Feb 04 2017

A371742 a(n) = Sum_{k=0..floor(n/2)} binomial(3n-k,n-2k).

Original entry on oeis.org

1, 3, 16, 92, 551, 3380, 21065, 132771, 843944, 5399802, 34731776, 224361283, 1454557294, 9458829681, 61670895633, 403003997300, 2638776935215, 17308508054848, 113709379928689, 748069400432262, 4927608724973776, 32495826854732633, 214521754579553129

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A108081, A371743, A371744.
Cf. A066380, A371754.
Cf. A183160.

a(n) = sum(k=0, n\2, binomial(3*n-k, n-2*k));

a(n) = [x^n] 1/((1-x-x^2) * (1-x)^(2*n)).

a(n) ~ 3^(3*n + 3/2) / (5 * sqrt(Pi*n) * 2^(2*n)). - Vaclav Kotesovec, Apr 05 2024

A387085 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*n+1,k).

Original entry on oeis.org

1, 0, 4, 8, 36, 120, 456, 1680, 6340, 23960, 91224, 348656, 1337896, 5149872, 19877904, 76907808, 298176516, 1158168792, 4505865144, 17555689008, 68490100536, 267518448912, 1046041377264, 4094231982048, 16039426479336, 62887835652720, 246761907761776, 968943740083040

Offset: 0

Seiichi Manyama, Aug 16 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000302, A000984, A001700, A026641, A141223, A377011, A386957.
Cf. A226733, A226705, A226751.
Cf. A005810, A079589, A183160.

[&+[(-3)^(n-k) * Binomial(2*n+1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025

Table[Sum[(-3)^(n-k)*Binomial[2*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)

a(n) = sum(k=0, n, (-3)^(n-k)*binomial(2*n+1, k));

a(n) = [x^n] (1+x)^(2*n+1)/(1+3*x).
a(n) = [x^n] 1/((1-x)^(n+1) * (1+2*x)).
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} (-2)^k * binomial(2*n-k,n-k).
G.f.: 1/( 4*x - 1 + 2*sqrt(1 - 4*x) ).
G.f.: 1/(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: g^2/((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: B(x)^2/(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.
D-finite with recurrence 3*n*a(n) +2*(-4*n+3)*a(n-1) +8*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Aug 19 2025

A171822 Triangle T(n,k) = binomial(2n-k, k)binomial(n+k, 2*k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 30, 30, 1, 1, 70, 225, 70, 1, 1, 135, 980, 980, 135, 1, 1, 231, 3150, 7056, 3150, 231, 1, 1, 364, 8316, 34650, 34650, 8316, 364, 1, 1, 540, 19110, 132132, 245025, 132132, 19110, 540, 1, 1, 765, 39600, 420420, 1288287, 1288287, 420420, 39600, 765, 1

Offset: 0

Roger L. Bagula, Dec 19 2009

			Triangle begins as:
  1;
  1,    1;
  1,    9,     1;
  1,   30,    30,       1;
  1,   70,   225,      70,       1;
  1,  135,   980,     980,     135,       1;
  1,  231,  3150,    7056,    3150,     231,       1;
  1,  364,  8316,   34650,   34650,    8316,     364,       1;
  1,  540, 19110,  132132,  245025,  132132,   19110,     540,     1;
  1,  765, 39600,  420420, 1288287, 1288287,  420420,   39600,   765,    1;
  1, 1045, 75735, 1166880, 5465460, 9018009, 5465460, 1166880, 75735, 1045, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A054142, A085478, A183160.

[Binomial(2*n-k, k)*Binomial(n+k, 2*k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 22 2021

Table[Binomial[2*n-k, k]*Binomial[n+k, 2*k], {n,0,10}, {k,0,n}]//Flatten

flatten([[binomial(2*n-k, k)*binomial(n+k, 2*k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 22 2021

T(n, k) = binomial(2*n-k, k)*binomial(n+k, 2*k) = A054142(n, k)*A085478(n, k).

Sum_{k=0..n} T(n, k) = Hypergeometric 4F3([-n, -n, 1/2 -n, n+1], [1/2, 1, -2*n], 1) = A183160(n). - G. C. Greubel, Feb 22 2021

Edited by G. C. Greubel, Feb 22 2021

A226706 G.f.: 1 / sqrt(1 + 12xG(x)^4 - 16xG(x)^5) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

Original entry on oeis.org

1, 2, 22, 256, 3174, 40862, 539376, 7247448, 98684230, 1357638124, 18831752122, 262974273200, 3692853486768, 52102851020154, 738102882420440, 10492839572260176, 149623214762194182, 2139329701502229300, 30661862088900836964, 440404155129948147776

Offset: 0

Paul D. Hanna, Jun 15 2013

nonn

			G.f.: A(x) = 1 + 2*x + 22*x^2 + 256*x^3 + 3174*x^4 + 40862*x^5 +...
A related series is G(x) = 1 + x*G(x), which begins
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
where A(x) = 1/sqrt(1 + 12*x*G(x)^4 - 16*x*G(x)^5).

Cf. A226705, A183160, A002295.

{a(n)=local(G=1+x); for(i=0, n,G=1+x*G^6+x*O(x^n)); polcoeff(1/sqrt(1+12*x*G^4-16*x*G^5), n)}
for(n=0, 30, print1(a(n), ", "))

Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} C(3*n+2*k,n-k)*C(3*n-2*k,k).

Self-convolution equals A226705.

A184553 a(n) = Sum_{k=0..n} C(3n+k,n-k)*C(4n-k,k).

Original entry on oeis.org

1, 6, 79, 1158, 17851, 283246, 4579306, 75013112, 1240774907, 20677408080, 346638007264, 5839169781594, 98755770443674, 1675855850883818, 28520685212980872, 486589040917153648, 8319672542504635643

Offset: 0

Paul D. Hanna, Jan 16 2011

nonn

			G.f.: A(x) = 1 + 6*x + 79*x^2 + 1158*x^3 + 17851*x^4 + 283246*x^5 +...
A(x)^(1/2) = 1 + 3*x + 35*x^2 + 474*x^3 + 6891*x^4 + 104360*x^5 +...+ A184554(n)*x^n +...
Given triangle T(n,k) = C(4n-k,k), which begins:
1;
3, 1;
15, 7, 1;
84, 45, 11, 1;
495, 286, 91, 15, 1;
3003, 1820, 680, 153, 19, 1; ...
ILLUSTRATE formula a(n) = Sum_{k=0..n} T(n,k)*T(n,n-k):
a(2) = 79 = 15*1 + 7*7 + 1*15;
a(3) = 1158 = 84*1 + 45*11 + 11*45 + 1*84;
a(4) = 17851 = 495*1 + 286*15 + 91*91 + 15*286 + 1*495;
a(5) = 283246 = 3003*1 + 1820*19 + 680*153 + 153*680 + 19*1820 + 1*3003; ...

Cf. A184554, A183160.

Table[Sum[Binomial[3*n + k, n - k]*Binomial[4*n - k, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 05 2020 *)

{a(n)=sum(k=0, n, binomial(3*n+k, n-k)*binomial(4*n-k, k))}

Self-convolution of A184554.
From Vaclav Kotesovec, Oct 05 2020: (Start)
Recurrence: 4608*n*(2*n - 1)*(3*n - 2)*(3*n - 1)*(6*n - 5)*(6*n - 1)*(824272*n^5 - 5898332*n^4 + 16800434*n^3 - 23808019*n^2 + 16784457*n - 4709052)*a(n) = 8*(5425780331264*n^11 - 55105585740928*n^10 + 246537716167440*n^9 - 639474746248560*n^8 + 1064922708464172*n^7 - 1190925449132724*n^6 + 908552008954195*n^5 - 470324138422910*n^4 + 160844796771909*n^3 - 34319567939418*n^2 + 4065509174760*n - 199264665600)*a(n-1) + 7*(7*n - 12)*(7*n - 11)*(7*n - 10)*(7*n - 9)*(7*n - 8)*(7*n - 6)*(824272*n^5 - 1776972*n^4 + 1449826*n^3 - 553989*n^2 + 97753*n - 6240)*a(n-2).
a(n) ~ 7^(7*n + 3/2) / (sqrt(Pi*n) * 2^(6*n + 4) * 3^(6*n + 1/2)). (End)

A201635 Triangle formed by T(n,n) = (-1)^n*Sum_{j=0..n} C(-n,j), T(n,k) = Sum_{j=0..k} T(n-1,j) for k=0..n-1, and n>=0, read by rows.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 2, 4, 6, 1, 3, 7, 13, 22, 1, 4, 11, 24, 46, 80, 1, 5, 16, 40, 86, 166, 296, 1, 6, 22, 62, 148, 314, 610, 1106, 1, 7, 29, 91, 239, 553, 1163, 2269, 4166, 1, 8, 37, 128, 367, 920, 2083, 4352, 8518, 15792, 1, 9, 46, 174, 541, 1461, 3544, 7896

Offset: 0

Peter Luschny, Nov 14 2012

Notation: If a sequence id is starred then the offset and/or some terms are different. Starred terms indicate the variance.
Row sums: [A026641 ] [1,  1,  4,  13,  46,  166,   610]
--
T(j+2, 2) [A000124*] [1*, 2 , 4,   7,  11,  16,     22]
T(j+3, 3) [A003600*] [1*, 2*, 6,  13,  24,  40,     62]
--
T(j  , j) [A072547 ] [1,  0,  2,   6,  22,  80,    296]
T(j+1, j) [A026641 ] [1,  1,  4,  13,  46,  166,   610]
T(j+2, j) [A014300 ] [1,  2,  7,  24,  86,  314,  1163]
T(j+3, j) [A014301*] [1,  3, 11,  40, 148,  553,  2083]
T(j+4, j) [A172025 ] [1,  4, 16,  62, 239,  920,  3544]
T(j+5, j) [A172061 ] [1,  5, 22,  91, 367, 1461,  5776]
T(j+6, j) [A172062 ] [1,  6, 29, 128, 541, 2232,  9076]
T(j+7, j) [A172063 ] [1,  7, 37, 174, 771, 3300, 13820]
--
T(2j  ,j) [Central ] [1,  1,  7,  40, 239, 1461,  9076]
T(2j+1,j) [A183160 ] [1,  2, 11,  62, 367, 2232, 13820]
T(2j+2,j) [        ] [1,  3, 16,  91, 541, 3300, 20476]
T(2j+3,j) [A199033*] [1,  4, 22, 128, 771, 4744, 29618]

			Triangle begins as:
[n]|k->
[0] 1
[1] 1, 0
[2] 1, 1,  2
[3] 1, 2,  4,  6
[4] 1, 3,  7, 13,  22
[5] 1, 4, 11, 24,  46,  80
[6] 1, 5, 16, 40,  86, 166, 296
[7] 1, 6, 22, 62, 148, 314, 610, 1106.

G. C. Greubel, Rows n=0..100 of triangle, flattened

A201635 := proc(n,k) option remember; local j;
if n=k then (-1)^n*add(binomial(-n,j), j=0..n)
else add(A201635(n-1,j), j=0..k) fi end:
for n from 0 to 7 do seq(A(n,k), k=0..n) od;

T[n_, k_]:= T[n, k]= If[k==n, (-1)^n*Sum[Binomial[-n, j], {j, 0, n}], Sum[T[n-1, j], {j, 0, k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 27 2019 *)

{T(n,k) = if(k==n, (-1)^n*sum(j=0,n, binomial(-n,j)), sum(j=0,k, T(n-1,j)))};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 27 2019

@CachedFunction
def A201635(n, k):
    if n==k: return (-1)^n*add(binomial(-n, j) for j in (0..n))
    return add(A201635(n-1, j) for j in (0..k))
for n in (0..7) : [A201635(n, k) for k in (0..n)]

cp's OEIS Frontend

A371770 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-3*k-1,n-3*k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

A199033 Number of ways to place n non-attacking bishops on a 2 X 2n board.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

PARI

Sage

Formula

Extensions

A225006 Number of n X n 0..1 arrays with rows unimodal and columns nondecreasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A371742 a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-k,n-2*k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A387085 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*n+1,k).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A171822 Triangle T(n,k) = binomial(2*n-k, k)*binomial(n+k, 2*k), read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A226706 G.f.: 1 / sqrt(1 + 12*x*G(x)^4 - 16*x*G(x)^5) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A371770 a(n) = Sum_{k=0..floor(n/3)} binomial(3n-3k-1,n-3*k).

A371742 a(n) = Sum_{k=0..floor(n/2)} binomial(3n-k,n-2k).

A171822 Triangle T(n,k) = binomial(2n-k, k)binomial(n+k, 2*k), read by rows.

A226706 G.f.: 1 / sqrt(1 + 12xG(x)^4 - 16xG(x)^5) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.