A060545 -id:A060545 - cp's OEIS frontend

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

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

A177784 a(n) = binomial(n^2, n) / ( n*(n+1) ).

Original entry on oeis.org

1, 7, 91, 1771, 46376, 1533939, 61474519, 2898753715, 157366449604, 9672348219898, 664226242466073, 50419551102990876, 4193002458968329488, 379189865879906158731, 37054233830964389244975

Offset: 2

Michel Lagneau, May 13 2010

nonn

All terms are integer because n and n+1 divide the binomial (cf. A060545, A177234).

Empirical: In the ring of symmetric functions over the fraction field Q(q, t), letting s(n) denote the Schur function indexed by n, a(n)*(-1)^(n+1) is equal to the coefficient of s(n) in nabla^(n)s(n) with q=t=1, where nabla denotes the "nabla operator" on symmetric functions. - John M. Campbell, Nov 18 2017

			For n = 3, binomial(9,3)/(3*4) =84/12 = 7.
For example, the coefficient of s(3) in nabla(nabla(nabla(s(3)))) is equal to q^6*t^2+q^5*t^3+q^4*t^4+q^3*t^5+q^2*t^6+q^4*t^3+q^3*t^4, and if we let q and t be equal to 1, this coefficient reduces to 7 = a(3). - _John M. Campbell_, Nov 18 2017

G. C. Greubel, Table of n, a(n) for n = 2..335

Cf. A060545, A177234.

[Binomial(n^2,n)/(2*Binomial(n+1,2)): n in [2..30]]; // G. C. Greubel, Jul 18 2024

A177784 := proc(n)
        binomial(n^2,n)/(n^2+n) ;
end proc:
seq(A177784(n),n=2..20) ; # R. J. Mathar, Nov 07 2011

Table[Binomial[n^2,n]/(2*Binomial[n+1,2]), {n,2,30}] (* G. C. Greubel, Jul 18 2024 *)

[binomial(n^2,n)//(n*(n+1)) for n in range(2,31)] # G. C. Greubel, Jul 18 2024

A177456 a(n) = binomial(n^2,n+1)/n.

Original entry on oeis.org

2, 42, 1092, 35420, 1391280, 64425438, 3442573064, 208710267480, 14162980464360, 1063958304188780, 87677864005521636, 7865449972066576656, 763126447532235966816, 79629871834780293333510

Offset: 2

Michel Lagneau, May 09 2010

nonn

n divides binomial(n^2,n+1).
Proof 1 :(n+1)*binomial(n^2,n+1) = n*(n-1)*binomial(n^2,n) => n divide binomial(n^2,n+1) because gcd(n,n+1) = 1.
Proof 2 : a(n) = binomial(n^2,n+1)/n = (n-1)*binomial(n^2-2,n-1)=> a(n) is an integer. - Michel Lagneau, May 13 2010

			For n=4, 1092 is in the sequence because binomial(16,5)/4 = 4368/4 = 1092.

G. C. Greubel, Table of n, a(n) for n = 2..335
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].
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. A000984, A014062, A060545, A177234, A177784, A177788.

[Binomial(n^2,n+1)/n: n in [2..30]]; // G. C. Greubel, Apr 29 2024

with(numtheory):n0:=30:T:=array(1..n0-1):for n from 2 to n0 do:T[n-1]:= (binomial(n*n,n+1))/n:od:print(T):

Table[Binomial[n^2,n+1]/n, {n,2,30}] (* G. C. Greubel, Apr 29 2024 *)

[binomial(n^2,n+1)/n for n in range(2,31)] # G. C. Greubel, Apr 29 2024

a(n) = binomial(n^2,n+1)/n.
From G. C. Greubel, Apr 29 2024: (Start)
a(n) = (n-1)*A177234(n).
a(n) = (n-1)*A177788(n)/n.
a(n) = n*(n-1)*A177784(n).
a(n) = A014062(n)/n. (End)

A177788 a(n) = binomial(n^2, n+1)/(n-1).

Original entry on oeis.org

4, 63, 1456, 44275, 1669536, 75163011, 3934369216, 234799050915, 15736644960400, 1170354134607658, 95648578915114512, 8520904136405458044, 821828481957792579648, 85317719822978885714475, 9485883860726883646713600, 1124586875214241546178986915

Offset: 2

Michel Lagneau, May 13 2010

nonn

The entries are integer for n >= 2 because binomial(n^2,n+1)/(n-1) = n*binomial(n^2-2,n-1), which is a product of two integers.

G. C. Greubel, Table of n, a(n) for n = 2..335

Cf. A000108, A014062, A060545, A177234, A177456, A177784.

[Binomial(n^2,n+1)/(n-1): n in [2..30]]; // G. C. Greubel, Apr 28 2024

n0:=30: T:=array(1..n0): T:=array(1..n0-1): for n from 2 to n0 do: T[n-1]:= (binomial(n^2,n+1))/(n-1): od: print(T):

Table[Binomial[n^2,n+1]/(n-1), {n,2,40}] (* G. C. Greubel, Apr 28 2024 *)

a(n) = binomial(n^2, n+1)/(n-1) \\ Charles R Greathouse IV, May 01 2024

[binomial(n^2,n+1)/(n-1) for n in range(2,31)] # G. C. Greubel, Apr 28 2024

a(n) = binomial(n^2,n+1)/(n-1).
a(n) = n * A177234(n).
a(n) = n^2 * A177784(n).

Removed redundant second Maple version - R. J. Mathar, May 14 2010

A177454 ( binomial(2*p,p) - 2)/p where p = prime(n).

Original entry on oeis.org

2, 6, 50, 490, 64130, 800046, 137270954, 1860277042, 357975249026, 1036802293087622, 15013817846943906, 47192717955016924590, 10360599532897359064118, 154361699651715243559786

Offset: 1

Michel Lagneau, May 09 2010

nonn

All entries are integer because binomial(2p, p) == 2 (mod p). [Proof: p!*binomial(2p, p) = 2p(2p - 1)(2p - 2) ... (p + 1) .
Therefore (p - 1)!*binomial(2p, p) = 2(2p - 1) ... (p + 1) == 2(p - 1)! (mod p).
Since p is prime: (p - 1)! <> 0 (mod p). Because Z/pZ is a finite field, we conclude that binomial(2p, p) == 2 (mod p).]

			a(1) = 2 because prime(1) = 2 and (binomial(4, 2) - 2)/2 = (6 - 2)/2 = 2.
a(4) = 490 because prime(4) = 7 and (binomial(14, 7) - 2)/7 = (3432 - 2)/7 = 490.

Amiram Eldar, Table of n, a(n) for n = 1..263
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5.

Cf. A000984, A060842, A060545, A024498.

[(Binomial(2*p,p)-2)/p where p is NthPrime(n):n in [1..14]]; // Marius A. Burtea, Aug 11 2019

with(numtheory): n0:=20: T:=array(1..n0): k:=1: for n from 1 to 72 do:if type(n,prime)=true then T[k]:= (binomial(2*n,n)-2)/n: k:=k+1: fi: od: print(T):

Table[(Binomial[2Prime[n], Prime[n]] - 2)/Prime[n], {n, 15}] (* Alonso del Arte, Feb 27 2013 *)

a(n) = (A000984(p) - 2) / p with p = A000040(n).

A303143 Number of minimum total dominating sets in the n-transposition graph.

Original entry on oeis.org

0, 1, 9, 784, 3686400

Offset: 1

Eric W. Weisstein, Apr 19 2018

a(n) matches all 4 terms of the finite sequence A168257(n).
a(n) matches A060545(n-1)^2 for n=2 to 4.
All terms are squares (see comment in A303146). - Andrew Howroyd, Jun 11 2025

Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
Eric Weisstein's World of Mathematics, Transposition Graph.

Cf. A303146, A303150 (set size), A321618.

a(5) from Christian Sievers, Jun 25 2025

A108288 Main diagonal of table A060543; a(n) = C((n+1)^2-1, n*(n+1)).

Original entry on oeis.org

1, 3, 28, 455, 10626, 324632, 12271512, 553270671, 28987537150, 1731030945644, 116068178638776, 8634941152058949, 705873715441872264, 62895036884524942320, 6067037854078498539696, 629921975126394617164575, 70043473196734767582082230

Offset: 0

Paul D. Hanna, May 31 2005

nonn

Cf. A060545, A108267, A060543, A108289.

PARI
```
a(n)=binomial((n+1)^2-1,n*(n+1))
```

a(n) = A060545(n+1). - R. J. Mathar, Aug 24 2008

A177466 a(n) = binomial(n^3, n^2) / (n^2 + n + 1).

Original entry on oeis.org

10, 360525, 23263187479980, 4195317468983232014706855, 3118254010126197540790713959812283024388, 13329519847131745416659896296893907619682838146506167497550

Offset: 2

Michel Lagneau, May 09 2010

nonn

All entries are integers. [Proof: binomial(n^3, n^2) / (n^2 + n + 1) = n^3 (n^3 - 1) (n^3 - 2)*...*(n^3- n^2 +1) / ( (n^2)! *(n^2 + n + 1)). With n^3 - 1 = (n-1)*(n^2 + n + 1), we obtain a(n) = n* binomial(n^3-2, n^2-2) / (n+1). Finally: (n+1) * binomial(n^3, n^2) * 1/ (n^2 + n + 1) = n*binomial(n^3-2, n^2-2). QED]
The step after "finally" seems to demonstrate merely that (n+1)*a(n) is an integer, but not that a(n) is itself an integer. Is the proof incomplete? - R. J. Mathar, Dec 06 2010
So far all that has been shown is that (n+1)*a(n) is an integer. To complete the proof, note that a(n) = n^3*(n-1)*(n^3-2)*...*(n^3-n^2) / (n^2*(n^2-1)!*(n^3-n^2)) = binomial(n^3-2,n^2-1)/n. Hence n*a(n) is also an integer, and so (n+1)*a(n) - n*a(n) = a(n) is an integer. Q.E.D. - N. J. A. Sloane, Dec 09 2010

			For n = 2, a(2) = binomial(8,4)/7 = 70/7 = 10.

G. C. Greubel, Table of n, a(n) for n = 2..23
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. A000108, A000984, A060545, A177234, A177456.

[Binomial(n^3,n^2)/(n^2+n+1): n in [2..12]]; // G. C. Greubel, Jul 18 2024

A177466 := proc(n) binomial(n^3,n^2)/(n^2+n+1); end proc:
seq(A177466(n),n=2..10) ; # R. J. Mathar, Dec 06 2010

Table[Binomial[n^3,n^2]/(n^2+n+1),{n,2,7}] (* Harvey P. Dale, Jan 24 2019 *)

[binomial(n^3,n^2)/(n^2+n+1) for n in range(2,13)] # G. C. Greubel, Jul 18 2024

cp's OEIS Frontend

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

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A177784 a(n) = binomial(n^2, n) / ( n*(n+1) ).

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A177456 a(n) = binomial(n^2,n+1)/n.

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A177788 a(n) = binomial(n^2, n+1)/(n-1).

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Magma

Maple

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A177454 ( binomial(2*p,p) - 2)/p where p = prime(n).

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Magma

Maple

Mathematica

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A303143 Number of minimum total dominating sets in the n-transposition graph.

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Crossrefs

Extensions

A108288 Main diagonal of table A060543; a(n) = C((n+1)^2-1, n*(n+1)).

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