A060539 -id:A060539 - cp's OEIS frontend

A005810 a(n) = binomial(4n,n).

1, 4, 28, 220, 1820, 15504, 134596, 1184040, 10518300, 94143280, 847660528, 7669339132, 69668534468, 635013559600, 5804731963800, 53194089192720, 488526937079580, 4495151581425648, 41432089765583440, 382460951663844400

Offset: 0

N. J. A. Sloane

Start off with 0 balls in a box. Find the number of ways you can throw 3 balls back out. Then continue to throw 4 balls into the box after each stage. (I.e., the first stage is 0. Then at the next stage there are 4 ways to throw 3 balls back out.) - Ruppi Rana (ruppirana007(AT)hotmail.com), Mar 03 2004
Central coefficients of A094527. - Paul Barry, Mar 08 2011
This is the case m = 2n in Catalan's formula (2m)!*(2n)!/(m!*(m+n)!*n!) - see Umberto Scarpis in References. - Bruno Berselli, Apr 27 2012
A generating function in terms of a (labyrinthine) solution to a depressed quartic equation is given in the Copeland link for signed A005810. - Tom Copeland, Oct 10 2012
Conjecture: a(n) == 4 (mod n^3) iff n is prime. - Gary Detlefs, Apr 03 2013
For prime p, the congruence a(p) = binomial(4*p,p) = 4 (mod p^3)  is a known generalization of Wolstenholme's theorem. See Mestrovic, Section 6, equation 35. - Peter Bala, Dec 28 2014

			G.f. = 1 + 4*x + 28*x^2 + 220*x^3 + 1820*x^4 + 15504*x^5 + 134596*x^6 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third Edition), page 11.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe, terms 101..213 from Muniru A Asiru)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Tom Copeland, Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers, 2012, pp. 5-6.
Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Ruppi Rana, Title? [Broken link]

Row 4 of A060539.
binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).
Cf. A007318, A182400, A262261, A378806, A378807.

List([0..20],n->Binomial(4*n,n)); # Muniru A Asiru, Mar 19 2018

a005810 n = a007318 (4*n) n  -- Reinhard Zumkeller, Mar 04 2012

[ Binomial(4*n,n): n in [0..100] ]; // Vincenzo Librandi, Apr 13 2011

seq(binomial(4*n,n),n=0..20); # Muniru A Asiru, Mar 19 2018

Table[Binomial[4n,n],{n,0,19}] (* Geoffrey Critzer, Sep 15 2013 *)

a(n) = binomial(4*n, n); \\ Altug Alkan, Mar 19 2018

from math import comb
def A005810(n): return comb(n<<2,n) # Chai Wah Wu, Aug 01 2023

a(n) is asymptotic to c*(256/27)^n/sqrt(n) with c = sqrt(2 / (3 Pi)) = 0.460658865961780639... - Benoit Cloitre, Jan 26 2003; corrected by Charles R Greathouse IV, Dec 14 2006
a(n) = Sum_{k=0..2n} binomial(2n,k) * binomial(2n,k-n). - Paul Barry, Mar 08 2011
G.f.: g/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011
D-finite with recurrence: 3*n*(3*n-1)*(3*n-2)*a(n) - 8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 02 2012
a(n) = binomial(4*n,n-1)*(3*n+1)/n. - Gary Detlefs, Apr 03 2013
a(n) = C(4*n-1,n-1)*C(16*n^2,2)/(3*n*C(4*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014
a(n) = Sum_{i,j,k = 0..n} binomial(n,i)*binomial(n,j)*binomial(n,k)* binomial(n,i+j+k). - Peter Bala, Dec 28 2014
a(n) = GegenbauerC(n, -2*n, -1). - Peter Luschny, May 07 2016
From Ilya Gutkovskiy, Nov 22 2016: (Start)
O.g.f.: 3F2(1/4,1/2,3/4; 1/3,2/3; 256*x/27).
E.g.f.: 3F3(1/4,1/2,3/4; 1/3,2/3,1; 256*x/27). (End)
a(n) = hypergeom([-3*n, -1*n], [1], 1). - Peter Luschny, Mar 19 2018
RHS of the identity Sum_{k = 0..2*n} (-1)^(n+k)*binomial(4*n, k)* binomial(4*n, 2*n-k) = binomial(4*n,n). - Peter Bala, Oct 07 2021
From Peter Bala, Feb 20 2022: (Start)
The o.g.f. A(x) satisfies the differential equation
(-256*x^3 + 27*x^2)*A(x)''' + (-1152*x^2 + 54*x)*A(x)'' + (-816*x + 6)*A(x)' - 24*A(x) = 0 with A(0) = 1, A'(0) = 4 and A''(0) = 56.
Algebraic equation: (1 - A(x))*(1 + 3*A(x))^3 +  256*x*A(x)^4 = 0.
Sum_{n >= 1} a(n)*( x*(3*x + 4)^3/(256*(1 + x)^4) )^n = x. (End)
From Amiram Eldar, Dec 07 2024: (Start)
Sum_{n>=1} 1/a(n) = A378806.
Sum_{n>=1} (-1)^n/a(n) = A378807. (End)
From Peter Bala, Jun 29 2025: (Start)
a(n) = (1/8)^n * Sum_{k = n..4*n} binomial(k, n) * binomial(4*n, k).
Sum_{n >= 0 } a(n)*(1/128)^n = (1/5)*(sqrt(2) + sqrt(7 + 5*sqrt(2))). (End)
From Seiichi Manyama, Aug 16 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(4*n+1,k).
G.f.: 1/(1 - 4*x*g^3) where g = 1+x*g^4 is the g.f. of A002293. (End)

More terms from Henry Bottomley, Oct 06 2000

Corrected by T. D. Noe, Jan 16 2007

A014062 a(n) = binomial(n^2, n).

Original entry on oeis.org

1, 1, 6, 84, 1820, 53130, 1947792, 85900584, 4426165368, 260887834350, 17310309456440, 1276749965026536, 103619293824707388, 9176358300744339432, 880530516383349192480, 91005567811177478095440, 10078751602022313874633200, 1190739044344491048895397910

Offset: 0

N. J. A. Sloane

nonn

Roberts states that Gupta and Khare show that a(n) > A002110(n) for 2 < n < 1794 and that a(n) < A002110(n) for n >= 1794, where A002110(n) is the product of the first n primes. - T. D. Noe, Oct 03 2007
This sequence describes the number of ways to arrange n objects in an n X n array (for example, stars in a flag's field pattern). - Tom Young (mcgreg265(AT)msn.com), Jun 17 2010
It appears that a(n) == n (mod n^3) only if n is 1, an odd prime, the square of an odd prime, or the cube of an odd prime. - Gary Detlefs, Aug 06 2013; corrected by Michel Marcus, May 29 2015

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 265.

T. D. Noe, Table of n, a(n) for n=0..100
Horst A. Alzer and Jozsef J. B. Šandor, On a binomial coefficient and a product of prime numbers, Appl. An. Disc. Math. 5 (2011) 87-92.
Harlan J. Brothers, Pascal's Prism: Supplementary Material.
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
Hansraj Gupta and S. P. Khare, On C(k^2,k) and the product of the first k primes, Publ. Fac. Electrotechn. Belgrade, Ser. Math. Phys. 25-29 (1977) 577-598.

Main diagonal of A060539.

Cf. A177234, A177456, A177784, A177788, A272094, A295773.

[Binomial(n^2,n): n in [0..30]]; // G. C. Greubel, Apr 29 2024

Table[Binomial[n^2,n],{n,0,22}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)
Table[SeriesCoefficient[(1+x)^(n^2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)

{a(n) = sum(k=0, n, binomial(n, k)*binomial(n^2-n, k))}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Nov 18 2015

[binomial(n^2,n) for n in range(31)] # G. C. Greubel, Apr 29 2024

a(n) ~ 1/sqrt(2*Pi) * (e*n)^(n - 1/2). - Charles R Greathouse IV, Jul 07 2007
a(n) = Sum_{k=0..n} binomial(n, k) * binomial(n^2 - n, k). - Paul D. Hanna, Nov 18 2015
a(n) = (n+1)*A177234(n). - R. J. Mathar, Jan 25 2019
From G. C. Greubel, Apr 29 2024: (Start)
a(n) = n*(n+1)*A177784(n).
a(n) = (n+1)*A177456(n)/(n-1).
a(n) = (n+1)*A177788(n)/n. (End)
a(n) = [x^n] (1+x)^(n^2). - Vaclav Kotesovec, Aug 06 2025

A004381 Binomial coefficient C(8n,n).

Original entry on oeis.org

1, 8, 120, 2024, 35960, 658008, 12271512, 231917400, 4426165368, 85113005120, 1646492110120, 32006008361808, 624668654531480, 12233149001721760, 240260199935164200, 4730523156632595024, 93343021201262177400, 1845382436487682488000

Offset: 0

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

T. D. Noe, Table of n, a(n) for n=0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Row 8 of A060539.

binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A005810 (k = 4), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A169958 - A169961 (k = 9 thru 12).

[Binomial(8*n, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014

Table[Binomial[8 n, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *)

from math import comb
def A004381(n): return comb(n<<3,n) # Chai Wah Wu, Aug 01 2023

a(n) = C(8*n-1,n-1)*C(64*n^2,2)/(3*n*C(8*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014
From Ilya Gutkovskiy, Jan 16 2017: (Start)
O.g.f.: 7F6(1/8,1/4,3/8,1/2,5/8,3/4,7/8; 1/7,2/7,3/7,4/7,5/7,6/7; 16777216*x/823543).
E.g.f.: 7F7(1/8,1/4,3/8,1/2,5/8,3/4,7/8; 1/7,2/7,3/7,4/7,5/7,6/7,1; 16777216*x/823543).
a(n) ~ 2^(24*n+1)/(sqrt(Pi*n)*7^(7*n+1/2)). (End)
From Peter Bala, Feb 20 2022: (Start)
The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 7*A(x))^7 +  (8^8)*x*A(x)^8 = 0.
Sum_{n >= 1} a(n)*( x*(7*x + 8)^7/(8^8*(1 + x)^8) )^n = x. (End)
From Seiichi Manyama, Aug 16 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(8*n+1,k).
G.f.: 1/(1 - 8*x*g^7) where g = 1+x*g^8 is the g.f. of A007556.
G.f.: g/(8-7*g) where g = 1+x*g^8 is the g.f. of A007556. (End)

A060543 Triangle, read by antidiagonals, where T(n,k) = C(n+nk+k, nk+k).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 10, 5, 1, 1, 35, 28, 7, 1, 1, 126, 165, 55, 9, 1, 1, 462, 1001, 455, 91, 11, 1, 1, 1716, 6188, 3876, 969, 136, 13, 1, 1, 6435, 38760, 33649, 10626, 1771, 190, 15, 1, 1, 24310, 245157, 296010, 118755, 23751, 2925, 253, 17, 1, 1, 92378, 1562275

Offset: 0

Henry Bottomley, Apr 02 2001

Main diagonal is A108288. Antidiagonal sums is A108289. Inverse binomial transforms of each row give triangle A108290. G.f. of row n multiplied by (1-x)^(n+1) equals g.f. of row n of triangle A108267 (rows sums of A108267 equal (n+1)^n).

			row 1: (2*n+1)/1!
row 2: (3*n+1)*(3*n+2)/2!
row 3: (4*n+1)*(4*n+2)*(4*n+3)/3!
row 4: (5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/4!
row 5: (6*n+1)*(6*n+2)*(6*n+3)*(6*n+4)*(6*n+5)/5!.
Table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
1,3,5,7,9,11,13,15,17,19,21,23,25,27,...
1,10,28,55,91,136,190,253,325,406,496,...
1,35,165,455,969,1771,2925,4495,6545,...
1,126,1001,3876,10626,23751,46376,82251,...
1,462,6188,33649,118755,324632,749398,...
1,1716,38760,296010,1344904,4496388,...

Harry J. Smith, Table of n, a(n) for n=0..252

Cf. A108290, A108267, A108288, A108289, A060544 (row 2), A015219 (row 3).

Rows include A000012, A001700, A025174. Columns include A000012, A005408, A060544. Main diagonal is A060545.

PARI
```
T(n,k)=binomial(n+n*k+k,n*k+k)
```

{ i=0; write("b060543.txt", "0 1"); for (m=0, 20, for (k=0, m + 1, n=m - k + 1; write("b060543.txt", i++, " ", binomial(n + n*k + k, n*k + k))); ) } \\ Harry J. Smith, Jul 06 2009

a(n) = A060539(n, k)/n = A007318(nk, k)/n = A060540(n, k)/A060540(n-1, k).

Entry revised by Paul D. Hanna, May 31 2005

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007

A169961 a(n) = binomial(12*n, n).

Original entry on oeis.org

1, 12, 276, 7140, 194580, 5461512, 156238908, 4529365776, 132601016340, 3911395881900, 116068178638776, 3461014728350400, 103619293824707388, 3112781199432937200, 93780365051563029360, 2832430653037446854640, 85733828145510955528212, 2600022926684976508835280

Offset: 0

N. J. A. Sloane, Aug 07 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Row 12 of A060539.

Cf. A000984, A005809, A005810, A001449, A004355, A004368, A004381, A169958, A169959, A169960.

[Binomial(12*n, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014

Table[Binomial[12 n, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *)
CoefficientList[Series[HypergeometricPFQ[Range[11]/12, Range[10]/11,(12^12)/(11^11)*x], {x,0,10}],x] (* Bradley Klee, Jul 01 2018 *)

a(n) = binomial(12*n, n); \\ Michel Marcus, Jul 02 2018

a(n) = C(12*n-1,n-1)*C(144*n^2,2)/(3*n*C(12*n+1,3)), n>0. -  Gary Detlefs, Jan 02 2014
From Bradley Klee, Jul 01 2018 : (Start)
G.f. G(x) and derivatives G^(n)(x)=d^n/dx^n G(x) satisfy a Picard-Fuchs type differential equation, 0=Sum_{m=0..11}(v1_{n}*x^(n+1)-v2_{n}*x^n)*G^(n)(x), with integer coefficient vectors:
v1={479001600, 647647046323200, 99278289544896000, 1290870365178240000, 4245175263164774400, 5313701967430348800, 3083267876011868160, 918801061774295040, 147161631039160320, 12624021804810240, 539424077119488, 8916100448256}
v2={0, 39916800, 14079254112000, 1273481816745600, 11475123393888000, 27687351298068000, 25909403608075680, 11200182937408080, 2427742942653600, 268452344620350, 14265583530550, 285311670611}
G.f.: G(x) = 11F10(m/12;n/11;12^12/11^11*x), m=1..11, n=1..10. (End)
From Vaclav Kotesovec, Jul 15 2018: (Start)
Recurrence: 11*n*(11*n - 10)*(11*n - 9)*(11*n - 8)*(11*n - 7)*(11*n - 6)*(11*n - 5)*(11*n - 4)*(11*n - 3)*(11*n - 2)*(11*n - 1)*a(n) = 41472*(2*n - 1)*(3*n - 2)*(3*n - 1)*(4*n - 3)*(4*n - 1)*(6*n - 5)*(6*n - 1)*(12*n - 11)*(12*n - 7)*(12*n - 5)*(12*n - 1)*a(n-1).
a(n) ~ 2^(24*n + 1/2) * 3^(12*n + 1/2) / (sqrt(Pi*n) * 11^(11*n + 1/2)). (End)
From Peter Bala, Feb 21 2022: (Start)
The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 11*A(x))^11 + (12^12)*x*A(x)^12 = 0.
Sum_{n >= 1} a(n)*( x*(11*x + 12)^11/(12^12*(1 + x)^12) )^n = x. (End)
From Seiichi Manyama, Aug 16 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(12*n+1,k).
G.f.: 1/(1 - 12*x*g^11) where g = 1+x*g^12.
G.f.: g/(12-11*g) where g = 1+x*g^12. (End)

A060538 Square array read by antidiagonals of number of ways of dividing n*k labeled items into n labeled boxes with k items in each box.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 20, 90, 24, 1, 70, 1680, 2520, 120, 1, 252, 34650, 369600, 113400, 720, 1, 924, 756756, 63063000, 168168000, 7484400, 5040, 1, 3432, 17153136, 11732745024, 305540235000, 137225088000, 681080400, 40320, 1, 12870

Offset: 1

Henry Bottomley, Apr 02 2001

			       1        1        1        1
       2        6       20       70
       6       90     1680    34650
      24     2520   369600 63063000

Harry J. Smith, Table of n, a(n) for n = 1..210

Subtable of A187783.
Rows include A000012, A000984, A006480, A008977, A008978 etc.
Columns include A000142, A000680, A014606, A014608, A014609 etc.
Main diagonal is A034841.

T(n,k)=(n*k)!/k!^n;
for(n=1, 6, for(k=1, 6, print1(T(n,k), ", ")); print) \\ Harry J. Smith, Jul 06 2009

T(n, k) = (nk)!/k!^n = T(n-1, k)*binomial(nk, k) = T(n-1, k)*A060539(n, k) = A060540(n, k)*A000142(k).

A060541 a(n) = binomial(4*n, 4).

Original entry on oeis.org

1, 70, 495, 1820, 4845, 10626, 20475, 35960, 58905, 91390, 135751, 194580, 270725, 367290, 487635, 635376, 814385, 1028790, 1282975, 1581580, 1929501, 2331890, 2794155, 3321960, 3921225, 4598126, 5359095, 6210820, 7160245, 8214570, 9381251, 10668000, 12082785

Offset: 1

Henry Bottomley, Apr 02 2001

Harry J. Smith, Table of n, a(n) for n=1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000027, A000332, A000384, A006566, A015219, A060539.

[Binomial(4 n, 4): n in [1..40]]; // Vincenzo Librandi, Jan 20 2015

Table[Binomial[4n, 4], {n, 100}] (* Wesley Ivan Hurt, Sep 27 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,70,495,1820,4845},40] (* Harvey P. Dale, Jan 13 2015 *)

a(n) = n*(2*n - 1)*(4*n - 1)*(4*n - 3)/3; \\ Harry J. Smith, Jul 06 2009

a(n) = n*(2n-1)*(4n-1)*(4n-3)/3.
a(n) = n * A015219(n-1) = A000332(4n) = A060539(n, 4).
G.f.: x*(1+65*x+155*x^2+35*x^3) / (1-x)^5. - R. J. Mathar, Oct 03 2011
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 6*log(2) - Pi.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*sqrt(2)*log(sqrt(2)-1) - log(2) + (2*sqrt(2) - 3/2)*Pi. (End)

Offset changed from 0 to 1 by Harry J. Smith, Jul 06 2009

A295772 a(n) = Sum_{k=0..n} binomial((n-k)*k, k).

Original entry on oeis.org

1, 1, 2, 4, 11, 41, 189, 1020, 6277, 43262, 328963, 2727076, 24425913, 234743744, 2406904525, 26202132494, 301579542517, 3656552470482, 46555182556971, 620695577790512, 8644238847922949, 125472134647552497, 1894393648378487895, 29696659293381522674

Offset: 0

Vaclav Kotesovec, Nov 27 2017

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 41*x^5 + 189*x^6 + 1020*x^7 + 6277*x^8 + 43262*x^9 + 328963*x^10 + ...

Robert Israel, Table of n, a(n) for n = 0..508

Cf. A060539, A226391, A264409.

seq(add(binomial((n-k)*k,k),k=0..n),n=0..30); # Robert Israel, Nov 27 2017

Table[Sum[Binomial[(n-k)*k, k], {k, 0, n}], {n, 0, 30}]

log(a(n)) ~ n*(log(n) - log(log(n)) + (log(log(n)) - 1)/log(n)). - Vaclav Kotesovec, Jan 10 2023

G.f. A(x) = 1 + x*Sum_{n>=0} x^n/n! * ( d^n/dy^n (1+y)^n/(1 - x*(1+y)^n) ) evaluated at y = 0. - Paul D. Hanna, Nov 13 2024

A323663 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is Sum_{j=1..n} binomial(j*k, k).

Original entry on oeis.org

1, 1, 3, 1, 7, 6, 1, 21, 22, 10, 1, 71, 105, 50, 15, 1, 253, 566, 325, 95, 21, 1, 925, 3256, 2386, 780, 161, 28, 1, 3433, 19489, 18760, 7231, 1596, 252, 36, 1, 12871, 119713, 154085, 71890, 17857, 2926, 372, 45, 1, 48621, 748342, 1303753, 747860, 214396, 38332, 4950, 525, 55

Offset: 1

Seiichi Manyama, Jan 23 2019

			Square array begins:
    1,   1,    1,     1,       1,        1, ...
    3,   7,   21,    71,     253,      925, ...
    6,  22,  105,   566,    3256,    19489, ...
   10,  50,  325,  2386,   18760,   154085, ...
   15,  95,  780,  7231,   71890,   747860, ...
   21, 161, 1596, 17857,  214396,  2695652, ...
   28, 252, 2926, 38332,  539028,  7941438, ...
   36, 372, 4950, 74292, 1197036, 20212950, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns 1-3 give A000217, A002412, A116689.
Rows 1-3 give A000012, A244174, A029848.
Main diagonal is A096131.
Cf. A060539.

A066704 Triangle with a(n,k) = C(n,floor(n/k)) with n>=k>=1.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 6, 4, 4, 1, 10, 5, 5, 5, 1, 20, 15, 6, 6, 6, 1, 35, 21, 7, 7, 7, 7, 1, 70, 28, 28, 8, 8, 8, 8, 1, 126, 84, 36, 9, 9, 9, 9, 9, 1, 252, 120, 45, 45, 10, 10, 10, 10, 10, 1, 462, 165, 55, 55, 11, 11, 11, 11, 11, 11, 1, 924, 495, 220, 66, 66, 12, 12, 12, 12, 12, 12

Offset: 1

Henry Bottomley, Jan 14 2002

			Rows start:
  1;
  1,  2;
  1,  3, 3;
  1,  6, 4, 4;
  1, 10, 5, 5, 5;
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Row sums are A051054.
Columns include (most of) A000012, A001405, A051033, A051036, A051052, A051053, A062947 etc.
n appears A008619 times in the n-th row.
Cf. A060539.

cp's OEIS Frontend

A005810 a(n) = binomial(4n,n).

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A014062 a(n) = binomial(n^2, n).

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A004381 Binomial coefficient C(8n,n).

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A060543 Triangle, read by antidiagonals, where T(n,k) = C(n+n*k+k, n*k+k).

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A169961 a(n) = binomial(12*n, n).

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A060538 Square array read by antidiagonals of number of ways of dividing n*k labeled items into n labeled boxes with k items in each box.

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A060543 Triangle, read by antidiagonals, where T(n,k) = C(n+nk+k, nk+k).