A051924 -id:A051924 - cp's OEIS frontend

A116647 Triangle of number of partitions that fit in an n X n box (but not in an (n-1) X (n-1) box) with Durfee square k.

1, 3, 1, 5, 8, 1, 7, 27, 15, 1, 9, 64, 84, 24, 1, 11, 125, 300, 200, 35, 1, 13, 216, 825, 1000, 405, 48, 1, 15, 343, 1911, 3675, 2695, 735, 63, 1, 17, 512, 3920, 10976, 12740, 6272, 1232, 80, 1, 19, 729, 7344, 28224, 47628, 37044, 13104, 1944, 99, 1, 21, 1000, 12825

Offset: 1

Franklin T. Adams-Watters, Feb 20 2006

			Triangle begins
   1;
   3,   1;
   5,   8,   1;
   7,  27,  15,   1;
   9,  64,  84,  24,   1;
  11, 125, 300, 200,  35,   1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Durfee Square.

Cf. A008459; row sums A051924.

Table[Binomial[n, k]^2 - Binomial[n - 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* G. C. Greubel, Nov 20 2017 *)

for(n=1,10, for(k=0,n, print1(binomial(n,k)^2 - binomial(n-1,k)^2, ", "))) \\ G. C. Greubel, Nov 20 2017

T(n,k) = binomial(n,k)^2 - binomial(n-1,k)^2.

A325228 Number of integer partitions of n such that the lesser of the maximum part and the number of parts is 3.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 6, 9, 13, 16, 20, 24, 28, 32, 38, 42, 48, 54, 60, 66, 74, 80, 88, 96, 104, 112, 122, 130, 140, 150, 160, 170, 182, 192, 204, 216, 228, 240, 254, 266, 280, 294, 308, 322, 338, 352, 368, 384, 400, 416, 434, 450, 468, 486, 504, 522, 542, 560

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

			The a(5) = 1 through a(10) = 16 partitions:
  (311)  (321)   (322)    (332)     (333)      (433)
         (411)   (331)    (422)     (432)      (442)
         (3111)  (421)    (431)     (441)      (532)
                 (511)    (521)     (522)      (541)
                 (3211)   (611)     (531)      (622)
                 (31111)  (3221)    (621)      (631)
                          (3311)    (711)      (721)
                          (32111)   (3222)     (811)
                          (311111)  (3321)     (3322)
                                    (32211)    (3331)
                                    (33111)    (32221)
                                    (321111)   (33211)
                                    (3111111)  (322111)
                                               (331111)
                                               (3211111)
                                               (31111111)

Column k = 3 of A325227.

Cf. A051924, A096771, A115720, A265283, A325224, A325225, A325229, A325231, A325232.

Table[Length[Select[IntegerPartitions[n],Min[Length[#],Max[#]]==3&]],{n,30}]

A337499 a(n) is the number of ballot sequences of length n tied or won by at most 2 votes.

Original entry on oeis.org

1, 2, 4, 6, 14, 20, 50, 70, 182, 252, 672, 924, 2508, 3432, 9438, 12870, 35750, 48620, 136136, 184756, 520676, 705432, 1998724, 2704156, 7696444, 10400600, 29716000, 40116600, 115000920, 155117520, 445962870

Offset: 0

Nachum Dershowitz, Aug 29 2020

Also the number of n-step walks on a path graph ending within 2 steps of the origin. Also the number of monotonic paths of length n ending within 2 steps of the diagonal.

Robert Israel, Table of n, a(n) for n = 0..3323

Cf. A001791, A128014.

Bisections give A000984 (odd part, starting from second element), A051924 (even part).

f:= gfun:-rectoproc({(4 + 4*n)*a(n) + (-12 - 4*n)*a(1 + n) + (-22 - 5*n)*a(2 + n) + (n + 4)*a(n + 3) + (6 + n)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 6},a(n),remember):
map(f, [$0..100]); # Robert Israel, Oct 08 2020

a(n) = A128014(n+1) + ((n+1) mod 2)*2*A001791(ceiling(n/2)).
D-finite with recurrence +(n+2)*a(n) +n*a(n-1) +(-5*n-2)*a(n-2) +4*(-n+1)*a(n-3) +4*(n-3)*a(n-4)=0. - Conjectured by R. J. Mathar, Sep 27 2020, verified by Robert Israel, Oct 08 2020
G.f.: ((4*x + 2)*sqrt(-4*x^2 + 1) + 4*x^2 + 4*x + 2)/(sqrt(-4*x^2 + 1)*(1 + sqrt(-4*x^2 + 1))^2). - Robert Israel, Oct 08 2020
a(n) ~ 2^(n - 1/2) * (5 + (-1)^n) / sqrt(Pi*n). - Vaclav Kotesovec, Mar 08 2023

A220956 (Binomial(2n, n) - binomial(2n - 2, n - 1)) (mod n^2) - n - 2.

Original entry on oeis.org

-3, -4, 0, -4, 0, 16, 0, 20, 18, 24, 0, -10, 0, 32, 28, 100, 0, 148, 0, 198, 403, 48, 0, 82, 250, 56, 18, 138, 0, 752, 0, 644, 436, 72, 705, 950, 0, 80, 369, 1178, 0, 1468, 0, 1322, 448, 96, 0, 1930, 1029, 1104, 766, 146, 0, 2488, 1680, 478, 3058, 120, 0, 2674, 0

Offset: 1

Gary Detlefs and Robert G. Wilson v, Feb 20 2013

Conjecture: a(n) = 0 iff n is an odd prime.
a(n) < 0 if n = 1, 2, 4, 12, 924, 1287, 2002, 2145, 3366, 3640, ... .
a(n) is odd if n = 1, 21, 35, 39, 49, 63, 69, 85, 91, 119, 123, ... .

			a(8)=20 since C(16,8) - C(14,7) (mod 64) = (12870 - 3432) (mod 64) = 9438 (mod 64) = 30 and 30 -8 -2 = 20.

Vincenzo Librandi, Table of n, a(n) for n = 1..10100

Cf. A051924, A000984.

[(Binomial(2*n,n)-Binomial(2*n-2,n-1)) mod n^2-n-2: n in [1..70]]; // Bruno Berselli, Feb 21 2013

f[n_] := Mod[Binomial[2 n, n] - Binomial[2 n - 2, n - 1], n^2] - n - 2; Array[f, 61]

a(n) = A051924(n) (mod n^2) -n -2.

A345013 Triangle read by rows, related to clusters of type D.

Original entry on oeis.org

1, 4, 3, 15, 20, 6, 56, 105, 60, 10, 210, 504, 420, 140, 15, 792, 2310, 2520, 1260, 280, 21, 3003, 10296, 13860, 9240, 3150, 504, 28, 11440, 45045, 72072, 60060, 27720, 6930, 840, 36

Offset: 1

F. Chapoton, Sep 30 2021

Let C_{n+1} be the cyclic quiver with n+1 vertices. Empirically, the n-th row is related to the green-mutation partial order on clusters for this quiver, restricted to clusters that do not meet the initial seed.
Apparently, value of the associated polynomials at -2 is A089849, up to sign.
By evaluating the associated polynomials at x-1, one apparently gets A062196.
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)*(x+n+2) / (n! * (n+2)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 31 2022
Chapoton's observation above is correct: the precise expansion is (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)*(x+n+2) / (n! * (n+2)!) = Sum_{k = 0..n} (-1)^k*T(n+1,k)*binomial(x+2*n+2-k, 2*n+2-k), as can be verified using the WZ algorithm. For example, n = 2 gives (x+1)^2*(x+2)^2*(x+3)*(x+4)/(2!*4!) = 15*binomial(x+6,6) - 20*binomial(x+5,5) + 6*binomial(x+4,4). - Peter Bala, Jun 24 2023

			Triangle begins:
[1] 1
[2] 4,    3
[3] 15,   20,    6
[4] 56,   105,   60,    10
[5] 210,  504,   420,   140,  15
[6] 792,  2310,  2520,  1260, 280,  21
[7] 3003, 10296, 13860, 9240, 3150, 504, 28
...

Cf. A001791 (T(n,1)), A000217 (T(n,n)), A026002 (row sums), A000012 (alternating row sum), A051924 (number of clusters of type D_n).

Cf. A089849, A062196, A063007, A253283.

row(n) = vector(n, k, k--; (n-k)*binomial(n,k)*binomial(2*n-k, n-1)/n); \\ Michel Marcus, Sep 30 2021

def T_row(n):
    return [(n-k)*binomial(n,k)*binomial(2*n-k,n-1)//n for k in range(n)]
for n in range(1, 8): print(T_row(n))

T(n, k) = (n-k)*binomial(n,k)*binomial(2*n-k, n-1)/n, for n >= 1 and 0 <= k < n.
From Peter Bala, Jun 24 2023: (Start)
As conjectured above by Chapoton we have
Sum_{k = 0..n-1} T(n,k)*(x - 1)^k = Sum_{k = 0..n-1} A062196(n-1,k)*x^k and
Sum_{k = 0..n-1} T(n,k)*(-2)^k = (-1)^floor(n/2)*A089849(n) for n >= 1 (both easily verified using the WZ algorithm). (End)

A241188 Triangle T(n,s) of Dynkin type D_n read by rows (n >= 2, 0 <= s <= n).

Original entry on oeis.org

1, 2, 1, 1, 3, 5, 5, 1, 4, 9, 16, 20, 1, 5, 14, 30, 55, 77, 1, 6, 20, 50, 105, 196, 294, 1, 7, 27, 77, 182, 378, 714, 1122, 1, 8, 35, 112, 294, 672, 1386, 2640, 4290, 1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 16445

Offset: 2

N. J. A. Sloane, Apr 24 2014

			Triangle begins:
1, 2, 1,
1, 3, 5, 5,
1, 4, 9, 16, 20,
1, 5, 14, 30, 55, 77,
1, 6, 20, 50, 105, 196, 294,
1, 7, 27, 77, 182, 378, 714, 1122,
1, 8, 35, 112, 294, 672, 1386, 2640, 4290,
1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 16445,
...

M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014.
M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, arXiv:1403.5827 [math.RT], 2014 and J. Int. Seq. 18 (2015) 15.10.6.

See A009766 for the case of type A.
See A059481 for the case of type B/C.
Diagonals give A029869, A051960, A029651, A051924. Row sums are also A051924.

f[t_, s_] := Binomial[t, s] (s + t)/t;
T[, 0] = 1; T[n, n_] := f[2 n - 2, n - 2]; T[n_, s_] := f[n + s - 2, s];
Table[T[n, s], {n, 2, 9}, {s, 0, n}] // Flatten (* Jean-François Alcover, Feb 12 2019 *)

T(n,s) = [n+s-2,s] for 0 <= s < n, T(n,n) = [2n-2,n-2], where [t,s] stands for binomial(t,s)*(s+t)/t.

A337500 a(n) is the number of ballot sequences of length n tied or won by at most 3 votes.

Original entry on oeis.org

1, 2, 4, 8, 14, 30, 50, 112, 182, 420, 672, 1584, 2508, 6006, 9438, 22880, 35750, 87516, 136136, 335920, 520676, 1293292, 1998724, 4992288, 7696444, 19315400, 29716000, 74884320, 115000920, 290845350, 445962870

Offset: 0

Nachum Dershowitz, Aug 30 2020

Also the number of n-step walks on a path graph ending within 3 steps of the origin.

Also the number of monotonic paths of length n ending within 3 steps of the diagonal.

Cf. A337499, A024483.

Bisections give A162551 (odd part, starting from second element), A051924 (even part).

a(n) = A337499(n) + (n mod 2)*A024483(floor((n+3)/2)).

Conjecture: D-finite with recurrence -(n+3)*(n-4)*a(n) +2*(n^2-2*n-11)*a(n-1) +4*(n-1)^2*a(n-2) -8*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 27 2020

A352184 Coxeter-Catalan numbers for the Coxeter groups A_0, A_1, A_2, A_3 = D_3, D_4, D_5, E_6, E_7, E_8.

Original entry on oeis.org

1, 2, 5, 14, 50, 182, 833, 4160, 25080

Offset: 0

N. J. A. Sloane, Mar 09 2022

Drew Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, Mem. Amer. Math. Soc. 202 (2009), no. 949, x+159. MR 2561274 16. See Table 2.8.
Drew Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, arXiv:math/0611106 [math.CO], 2006-2007.

Cf. A000108, A051924.

A378377 Triangle read by rows: T(n,k) is the number of non-descending sequences with length k such that the maximum of the length and the last number is n.

Original entry on oeis.org

1, 1, 3, 1, 3, 10, 1, 4, 10, 35, 1, 5, 15, 35, 126, 1, 6, 21, 56, 126, 462, 1, 7, 28, 84, 210, 462, 1716, 1, 8, 36, 120, 330, 792, 1716, 6435, 1, 9, 45, 165, 495, 1287, 3003, 6435, 24310, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 92378

Offset: 1

Zlatko Damijanic, Nov 24 2024

Also the T(n,k) is the number of integer partitions (of any positive integer) with length k such that the maximum of the length and the largest part is n.

When k < n, then the last number is n.

			Triangle begins:
  1
  1 3
  1 3 10
  1 4 10 35
  1 5 15 35 126
  1 6 21 56 126 462
  1 7 28 84 210 462 1716
  ...
For T(3,1) solution is |{(3)}| = 1.
For T(3,2) solution is |{(1,3), (2,3), (3,3)}| = 3.
For T(3,3) solution is |{(1,1,1), (1,1,2), (1,1,3), (1,2,2), (1,2,3), (1,3,3), (2,2,2), (2,2,3), (2,3,3), (3,3,3)}| = 10.

Cf. A051924 (row sums), A001700 (right diagonal).

T[n_, k_] := Which[
  k == 1, 1,
  k == n, Binomial[2n-1, n],
  k == n-1, T[n-1, n-1],
  1 < k < n-1, T[n-1, k] + T[n, k-1]
];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten

PARI
```
T(n,k)={if(kAndrew Howroyd, Nov 24 2024
```

T(n,n) = binomial(2*n-1,n).

T(n,k) = binomial(k+n-2, n-1) for k < n.

cp's OEIS Frontend

A116647 Triangle of number of partitions that fit in an n X n box (but not in an (n-1) X (n-1) box) with Durfee square k.

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A325228 Number of integer partitions of n such that the lesser of the maximum part and the number of parts is 3.

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A337499 a(n) is the number of ballot sequences of length n tied or won by at most 2 votes.

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Maple

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A220956 (Binomial(2n, n) - binomial(2n - 2, n - 1)) (mod n^2) - n - 2.

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Magma

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A345013 Triangle read by rows, related to clusters of type D.

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Sage

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A241188 Triangle T(n,s) of Dynkin type D_n read by rows (n >= 2, 0 <= s <= n).

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A337500 a(n) is the number of ballot sequences of length n tied or won by at most 3 votes.

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A352184 Coxeter-Catalan numbers for the Coxeter groups A_0, A_1, A_2, A_3 = D_3, D_4, D_5, E_6, E_7, E_8.

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A378377 Triangle read by rows: T(n,k) is the number of non-descending sequences with length k such that the maximum of the length and the last number is n.

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