A033486 -id:A033486 - cp's OEIS frontend

A033487 a(n) = n(n+1)(n+2)*(n+3)/4.

0, 6, 30, 90, 210, 420, 756, 1260, 1980, 2970, 4290, 6006, 8190, 10920, 14280, 18360, 23256, 29070, 35910, 43890, 53130, 63756, 75900, 89700, 105300, 122850, 142506, 164430, 188790, 215760, 245520, 278256, 314160, 353430, 396270, 442890, 493506, 548340, 607620

Offset: 0

N. J. A. Sloane

Non-vanishing diagonal of (A132440)^4/4. Third subdiagonal of unsigned A238363 without the zero. Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices of the complete graph K_4. - Tom Copeland, Apr 05 2014
Total number of pips on a set of trominoes (3-armed dominoes) with up to n pips on each arm. - Alan Shore and N. J. A. Sloane, Jan 06 2016
Also the number of minimum connected dominating sets in the (n+2)-crown graph. - Eric W. Weisstein, Jun 29 2017
Crossing number of the (n+3)-cocktail party graph (conjectured). - Eric W. Weisstein, Apr 29 2019
Sum of all numbers in ordered triples (x,y,z) where 0 <= x <= y <= z <= n. - Edward Krogius, Jul 31 2022

			G.f. = 6*x + 30*x^2 + 90*x^3 + 210*x^4 + 420*x^5 + 756*x^6 + 1260*x^7 + ...

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Vincenzo Librandi, Table of n, a(n) for n = 0..690
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq., Vol. 13 (2010), Article 10.4.4.
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Aleksandar Petojević and Nenad Đapić, The vAm(a,b,c;z) function, Preprint 2013.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Hasan Unal, Proof Without Words: Sums of Products of Three Consecutive Integers, Mathematics Magazine, Vol. 88, No. 1 (February 2015), pp. 37-38.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Connected Dominating Set.
Eric Weisstein's World of Mathematics, Crown Party Graph.
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Index entries for sequences related to Bessel functions or polynomials
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000332, A002378, A033486, A033488, A034827, A050534, A130534, A132440, A238363.
Partial sums of A007531.
A row of the array in A129533.
A column of the triangle in A331430.
Sequences of the form binomial(n+k,k)*binomial(n+k+2,k): A000012 (k=0), A005563 (k=1), this sequence (k=2), A027790 (k=3), A107395 (k=4), A107396 (k=5), A107397 (k=6), A107398 (k=7), A107399 (k=8).

[n*(n+1)*(n+2)*(n+3)/4: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

[seq(binomial(n+3,4)*6, n=0..40)]; # Zerinvary Lajos, Jul 18 2006

Table[Times @@ (n + Range[0, 3])/4, {n, 0, 40}] (* Harvey P. Dale, Nov 27 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 6, 30, 90, 210}, 40] (* Harvey P. Dale, Nov 27 2013 *)
Table[6 Binomial[n+3, 4], {n,0,40}] (* Eric W. Weisstein, Jun 29 2017 *)
Times @@@ Table[n+k, {n, 0, 40}, {k, 0, 3}]/4 (* Eric W. Weisstein, Apr 29 2019 *)

a(n)=6*binomial(n+3,4) \\ Charles R Greathouse IV, Apr 17 2012

concat(0, Vec(6*x/(1-x)^5 + O(x^100))) \\ Altug Alkan, Nov 29 2015

def A033487(n): return 6*binomial(n+3,4)
print([A033487(n) for n in range(41)]) # G. C. Greubel, Feb 08 2025

From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 10 2001: (Start)
G.f.: 6*x/(1-x)^5.
a(n) = 6*binomial(n+3, 4) = 6*A000332(n+3).
a(n) = a(n-1) + A007531(n+1).
a(n) = Sum_{i=0..n} i*(i+1)*(i+2). (End)
Constant term in Bessel polynomial {y_n(x)}''.
a(n) = binomial(n+1,2)*binomial(n+3,2) = A000217(n)*A000217(n+2). - Zerinvary Lajos, May 25 2005
a(n) = binomial(n+2,2)^2 - binomial(n+2,2). - Zerinvary Lajos, May 17 2006
From Zerinvary Lajos, May 11 2007: (Start)
a(n-1) = Sum_{j=1..n} Sum_{i=2..n} i*j.
a(n) = Sum_{j=1..n} j*(n+2)*(n-1)/2. (End)
Sum_{n>0} 1/a(n) = 2/9. - Enrique Pérez Herrero, Nov 10 2013
a(-3-n) = a(n) = 2 * binomial(binomial(n+2, 2), 2). - Michael Somos, Apr 06 2014
a(n) = A002378(binomial(n+2,2)-1). - Salvador Cerdá, Nov 04 2016
a(n) = Sum_{k=0..n} A007531(k+2). See Proof Without Words link. - Michel Marcus, Oct 29 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 16*log(2)/3 - 32/9. - Amiram Eldar, Nov 02 2021
E.g.f.: exp(x)*x*(24 + 36*x + 12*x^2 + x^3)/4. - Stefano Spezia, Jul 03 2025

A011915 a(n) = floor(n(n-1)(n-2)*(n-3)/5).

Original entry on oeis.org

0, 0, 0, 0, 4, 24, 72, 168, 336, 604, 1008, 1584, 2376, 3432, 4804, 6552, 8736, 11424, 14688, 18604, 23256, 28728, 35112, 42504, 51004, 60720, 71760, 84240, 98280, 114004, 131544, 151032, 172608, 196416, 222604, 251328, 282744, 317016, 354312

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1,-4,6,-4,1).

Sequences of the form floor(24*binomial(n,4)/m): A052762 (m=1), A033486 (m=2), A162668 (m=3), A033487 (m=4), this sequence (m=5), A033488 (m=6), A011917 (m=7), A050534 (m=8), A011919 (m=9), 2*A011930 (m=10), A011921 (m=11), A034827 (m=12), A011923 (m=13), A011924 (m=14), A011925 (m=15), A011926 (m=16), A011927 (m=17), A011928 (m=18), A011929 (m=19), A011930 (m=20), A011931 (m=21), A011932 (m=22), A011933 (m=23), A000332 (m=24), A011935 (m=25),A011936 (m=26), A011937 (m=27), A011938 (m=28), A011939 (m=29), A011940 (m=30), A011941 (m=31), A011942 (m=32), A011795 (m=120).

[Floor(n*(n-1)*(n-2)*(n-3)/5): n in [0..60]]; // Vincenzo Librandi, Jun 19 2012

Table[Floor[n(n-1)(n-2)(n-3)/5], {n,60}] (* Stefan Steinerberger, Apr 10 2006 *)
CoefficientList[Series[4*x^4*(1+2*x+2*x^3+x^4)/((1-x)^4*(1+x^5)),{x,0,60}],x] (* Vincenzo Librandi, Jun 19 2012 *)

[24*binomial(n,4)//5 for n in range(61)] # G. C. Greubel, Oct 20 2024

a(n) = +4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +a(n-5) -4*a(n-6) +6*a(n-7) -4*a(n-8) +a(n-9).
G.f.: 4*x^4*(1+2*x+2*x^3+x^4) / ( (1-x)^5*(1+x+x^2+x^3+x^4) ). - R. J. Mathar, Apr 15 2010
a(n) = 4*A011930(n). - G. C. Greubel, Oct 20 2024

More terms from Stefan Steinerberger, Apr 10 2006

Zero added in front by R. J. Mathar, Apr 15 2010

A144207 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph consists of a single node or has a unique cycle of length 3.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 4, 12, 1, 0, 0, 10, 60, 150, 1, 0, 0, 20, 180, 900, 2160, 1, 0, 0, 35, 420, 3150, 15180, 36015, 1, 0, 0, 56, 840, 8400, 60750, 291060, 688128, 1, 0, 0, 84, 1512, 18900, 182270, 1311240, 6300672, 14880348, 1, 0, 0, 120, 2520, 37800

Offset: 0

Alois P. Heinz, Sep 14 2008

			T(5,4) = 60 = 5*12, because there are 5 possibilities for a single node and T(4,4) = 12:
.1-2. .1-2. .1-2. .1.2. .1.2. .1-2. .1.2. .1.2. .1-2. .1-2. .1-2. .1-2.
.|X.. .|/|. .|/.. ..X|. .|/|. ../|. .|X.. .|\|. .|\.. ..X|. .|\|. ..\|.
.3.4. .3.4. .3-4. .3-4. .3-4. .3-4. .3-4. .3-4. .3-4. .3.4. .3.4. .3-4.
Triangle begins:
1;
1, 0;
1, 0, 0;
1, 0, 0, 1;
1, 0, 0, 4, 12;
1, 0, 0, 10, 60, 150;

Alois P. Heinz, Rows n = 0..140, flattened

Columns 0, 1+2, 3, 4 give: A000012, A000004, A000292, A033486 or A112415. Diagonal gives: A053507. Row sums give: A144208. Cf. A007318.

T:= proc(n,k) option remember; if k=0 then 1 elif k<0 or n

t[n_, k_] := t[n, k] = Which[k == 0, 1, k < 0 || n < k, 0, k == n, Binomial[n-1, 2]*n^(n-3), True, t[n-1, k] + Sum[Binomial[n-1, j]*t[j+1, j+1]*t[n-1-j, k-j-1], {j, 2, k-1}]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 11}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

T(n,0) = 1, T(n,k) = 0 if k<0 or n

A293617 Array of triangles read by ascending antidiagonals, T(m, n, k) = Pochhammer(m, k) * Stirling2(n + m, k + m) with m >= 0, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 2, 1, 0, 1, 10, 3, 7, 3, 0, 1, 15, 4, 25, 12, 2, 0, 1, 21, 5, 65, 30, 6, 1, 0, 1, 28, 6, 140, 60, 12, 15, 7, 0, 1, 36, 7, 266, 105, 20, 90, 50, 12, 0, 1, 45, 8, 462, 168, 30, 350, 195, 60, 6, 0, 1, 55, 9, 750, 252, 42, 1050, 560, 180, 24, 1, 0

Offset: 0

Peter Luschny, Oct 20 2017

			Array starts:
m\j| 0   1  2     3       4       5       6       7       8       9      10
---|-----------------------------------------------------------------------
m=0| 1,  0, 0,    0,      0,      0,      0,      0,      0,      0,      0
m=1| 1,  1, 1,    1,      3,      2,      1,      7,     12,      6,      1
m=2| 1,  3, 2,    7,     12,      6,     15,     50,     60,     24,     31
m=3| 1,  6, 3,   25,     30,     12,     90,    195,    180,     60,    301
m=4| 1, 10, 4,   65,     60,     20,    350,    560,    420,    120,   1701
m=5| 1, 15, 5,  140,    105,     30,   1050,   1330,    840,    210,   6951
m=6| 1, 21, 6,  266,    168,     42,   2646,   2772,   1512,    336,  22827
m=7| 1, 28, 7,  462,    252,     56,   5880,   5250,   2520,    504,  63987
m=8| 1, 36, 8,  750,    360,     72,  11880,   9240,   3960,    720, 159027
m=9| 1, 45, 9, 1155,    495,     90,  22275,  15345,   5940,    990, 359502
   A000217, A001296,A027480,A002378,A001297,A293475,A033486,A007531,A001298
.
m\j| ...      11      12      13      14
---|-----------------------------------------
m=0| ...,      0,      0,      0,      0, ... [A000007]
m=1| ...,     15,     50,     60,     24, ... [A028246]
m=2| ...,    180,    390,    360,    120, ... [A053440]
m=3| ...,   1050,   1680,   1260,    360, ... [A294032]
m=4| ...,   4200,   5320,   3360,    840, ...
m=5| ...,  13230,  13860,   7560,   1680, ...
m=6| ...,  35280,  31500,  15120,   3024, ...
m=7| ...,  83160,  64680,  27720,   5040, ...
m=8| ..., 178200, 122760,  47520,   7920, ...
m=9| ..., 353925, 218790,  77220,  11880, ...
         A293476,A293608,A293615,A052762, ...
.
The parameter m runs over the triangles and j indexes the triangles by reading them by rows. Let T(m, n) denote the row [T(m, n, k) for 0 <= k <= n] and T(m) denote the triangle [T(m, n) for n >= 0]. Then for instance T(2) is the triangle A053440, T(3, 2) is row 2 of A294032 (which is [25, 30, 12]) and T(3, 2, 1) = 30.
.
Remark: To adapt the sequences A028246 and A053440 to our enumeration use the exponential generating functions exp(x)/(1 - y*(exp(x) - 1)) and exp(x)*(2*exp(x) - y*exp(2*x) + 2*y*exp(x) - 1 - y)/(1 - y*(exp(x) - 1))^2 instead of those indicated in their respective entries.

Eric Weisstein's World of Mathematics, Nørlund Polynomial.

A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A027480(n) = T(n, 2, 1),
A002378(n) = T(n, 2, 2), A001297(n) = T(n, 3, 0), A293475(n) = T(n, 3, 1),
A033486(n) = T(n, 3, 2), A007531(n) = T(n, 3, 3), A001298(n) = T(n, 4, 0),
A293476(n) = T(n, 4, 1), A293608(n) = T(n, 4, 2), A293615(n) = T(n, 4, 3),
A052762(n) = T(n, 4, 4), A052787(n) = T(n, 5, 5), A000225(n) = T(1, n, 1),
A028243(n) = T(1, n, 2), A028244(n) = T(1, n, 3), A028245(n) = T(1, n, 4),
A032180(n) = T(1, n, 5), A228909(n) = T(1, n, 6), A228910(n) = T(1, n, 7),
A000225(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
A028246(n,k) = T(1, n, k), A053440(n,k) = T(2, n, k), A294032(n,k) = T(3, n, k),
A293926(n,k) = T(n, n, k), A124320(n,k) = T(n, k, k), A156991(n,k) = T(k, n, n).
Cf. A293616.

A293617 := proc(m, n, k) option remember:
if m = 0 then 0^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
(k+m)*A293617(m,n-1,k) + k*A293617(m,n-1,k-1) + A293617(m-1,n,k) fi end:
for m in [$0..4] do for n in [$0..6] do print(seq(A293617(m, n, k), k=0..n)) od od;
# Sample uses:
A027480 := n -> A293617(n, 2, 1): A293608 := n -> A293617(n, 4, 2):
# Flatten:
a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
seq(A293617(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);

T[m_, n_, k_] := Pochhammer[m, k] StirlingS2[n + m, k + m];
For[m = 0, m < 7, m++, Print[Table[T[m, n, k], {n,0,6}, {k,0,n}]]]
A293617Row[m_, n_] := Table[T[m, n, k], {k,0,n}];
(* Sample use: *)
A293926Row[n_] := A293617Row[n, n];

T(m,n,k) = (k + m)*T(m, n-1, k) + k*T(m, n-1, k-1) + T(m-1, n, k) with boundary conditions T(0, n, k) = 0^n; T(m, n, k) = 0 if k<0 or k>n; and T(m, 0, k) = 0^k.

T(m,n,k) = Pochhammer(m, k)*binomial(n + m, k + m)*NorlundPolynomial(n - k, -k - m).

A342589 T(n,k) is the number of posets of n labeled elements with k covering relations (n>=1, k>=0). Triangle read by rows.

Original entry on oeis.org

1, 1, 2, 1, 6, 12, 1, 12, 60, 128, 18, 1, 20, 180, 880, 2090, 960, 100, 1, 30, 420, 3480, 17550, 47772, 43920, 15000, 1710, 140, 1, 42, 840, 10360, 84630, 452004, 1428868, 2094960, 1465170, 491540, 90594, 10080, 770

Offset: 1

R. J. Mathar, Mar 16 2021

			The triangle starts:
  1: 1
  2: 1 2
  3: 1 6 12
  4: 1 12 60 128 18
  5: 1 20 180 880 2090 960 100
  6: 1 30 420 3480 17550 47772 43920 15000 1710 140
  7: 1 42 840 10360 84630 452004 1428868 2094960 1465170 491540 90594 10080 770

Brendan McKay, Table of T(n,k) for n = 1..13
Index entries for sequences related to posets

Cf. A001035 (row sums), A002378 (k=1), A033486 (k=2?), A342447 (unlabeled), A342588 (connected).

A112415 a(n) = C(1+n,1) * C(2+n,1) * C(4+n,2).

Original entry on oeis.org

12, 60, 180, 420, 840, 1512, 2520, 3960, 5940, 8580, 12012, 16380, 21840, 28560, 36720, 46512, 58140, 71820, 87780, 106260, 127512, 151800, 179400, 210600, 245700, 285012, 328860, 377580, 431520, 491040, 556512, 628320, 706860, 792540, 885780, 987012, 1096680

Offset: 0

Zerinvary Lajos, Dec 09 2005

			n=0: C(1+0,1)*C(2+0,1)*C(4+0,2) = C(1,1)*C(2,1)*C(4,2) = 1*2*6 = 12;
n=10: C(1+10,1)*C(2+10,1)*C(4+10,2) = C(11,1)*C(12,1)*C(14,2) = 11*12*91 = 12012.

Vincenzo Librandi, Table of n, a(n) for n = 0..680
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000332, A033486.

[(n+1)*(n+2)*(n+3)*(n+4)/2: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

Table[(n+1)(n+2)Binomial[4+n,2],{n,0,30}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{12,60,180,420,840},31] (* Harvey P. Dale, Jul 24 2011 *)

From R. J. Mathar, Aug 15 2008: (Start)
a(n) = (n+1)*(n+2)*(n+3)*(n+4)/2 = A033486(n+1) = 12*A000332(n+4).
O.g.f.: 12/(1-x)^5. (End)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=12, a(1)=60, a(2)=180, a(3)=420, a(4)=840. - Harvey P. Dale, Jul 24 2011
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=0} 1/a(n) = 1/9.
Sum_{n>=0} (-1)^n/a(n) = 8*(3*log(2)-2)/9. (End)

Showing 1-6 of 6 results.

cp's OEIS Frontend

A033487 a(n) = n*(n+1)*(n+2)*(n+3)/4.

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A011915 a(n) = floor(n*(n-1)*(n-2)*(n-3)/5).

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A144207 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph consists of a single node or has a unique cycle of length 3.

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A293617 Array of triangles read by ascending antidiagonals, T(m, n, k) = Pochhammer(m, k) * Stirling2(n + m, k + m) with m >= 0, n >= 0 and 0 <= k <= n.

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A342589 T(n,k) is the number of posets of n labeled elements with k covering relations (n>=1, k>=0). Triangle read by rows.

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A112415 a(n) = C(1+n,1) * C(2+n,1) * C(4+n,2).

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A033487 a(n) = n(n+1)(n+2)*(n+3)/4.

A011915 a(n) = floor(n(n-1)(n-2)*(n-3)/5).