A107396 -id:A107396 - 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

A062196 Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).

Original entry on oeis.org

1, 1, 3, 1, 8, 6, 1, 15, 30, 10, 1, 24, 90, 80, 15, 1, 35, 210, 350, 175, 21, 1, 48, 420, 1120, 1050, 336, 28, 1, 63, 756, 2940, 4410, 2646, 588, 36, 1, 80, 1260, 6720, 14700, 14112, 5880, 960, 45, 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55

Offset: 0

Wolfdieter Lang, Jun 19 2001

Also the coefficient triangle of certain polynomials N(2; m,x) := Sum_{k=0..m} T(m,k)*x^k. The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=2) Laguerre triangle L(2; n+m,m) = A062139(n+m,m), n >= 0, is N(2; m,x)/(1-x)^(3+2*m), with the row polynomials N(2; m,x).

			Triangle starts:
  n\k 0...1.....2......3..... 4.....;
  [0] 1;
  [1] 1,  3;
  [2] 1,  8,    6;
  [3] 1, 15,   30,    10;
  [4] 1, 24,   90,    80,    15;
  [5] 1, 35,  210,   350,   175,    21;
  [6] 1, 48,  420,  1120,  1050,   336,    28;
  [7] 1, 63,  756,  2940,  4410,  2646,   588,    36;
  [8] 1, 80, 1260,  6720, 14700, 14112,  5880,   960,   45;
  [9] 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), this sequence (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
Sums include: A001791 (row), (-1)^n*A089849(n+1) (alternating sign row).
Diagonals: A000217 (k=n), A002417 (k=n-1), A001297 (k=n-2), A105946 (k=n-3), A105947 (k=n-4), A105948 (k=n-5), A107319 (k=n-6).
Columns: A005563 (k=1), A033487 (k=2), A027790 (k=3), A107395 (k=4), A107396 (k=5), A107397 (k=6), A107398 (k=7), A107399 (k=8).
Cf. A000108, A089849.

A062196:= func;
[A062196(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 21 2025

T := (n, k) -> binomial(n, k)*binomial(n + 2, k);
seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, Sep 30 2021

A062196[n_, k_]:= Binomial[n, k]*Binomial[n+2, k];
Table[A062196[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 21 2025 *)

def A062196(n,k): return binomial(n,k)*binomial(n+2,k)
print(flatten([[A062196(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 21 2025

T(m, k) = [x^k] N(2; m, x), where N(2; m, x) = ((1-x)^(3+2*m))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+3))).
N(2; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+2-j)!/((m+2)!*(m-j)!)*(x^(m-j)))*(1-x)^j).
T(n,m) = binomial(n, m)*(binomial(n+1, m) + binomial(n+1, m-1)). - Vladimir Kruchinin, Apr 06 2018
From G. C. Greubel, Feb 21 2025: (Start)
T(2*n, n) = (n+1)^2*A000108(n)*A000108(n+1).
T(2*n-1, n) = (4*n^2 - 1)*A000108(n-1)*A000108(n), n >= 1.
T(2*n+1, n) = (1/2)*binomial(n+2,2)*A000108(n+1)*A000108(n+2). (End)

New name by Peter Luschny, Sep 30 2021

A133111 a(n) = (1/(1!2!3!4!))Sum_{1 <= x_1, x_2, x_3, x_4 <= n} |det V(x_1,x_2,x_3,x_4)|, where V(x_1,x_2,x_3,x_4) is the Vandermonde matrix of order 4.

Original entry on oeis.org

0, 0, 0, 1, 16, 126, 672, 2772, 9504, 28314, 75504, 184041, 416416, 884884, 1782144, 3426384, 6325632, 11267532, 19442016, 32605881, 53300016, 85131970, 133138720, 204246900, 307850400, 456528150, 666928080, 960846705, 1366537536, 1920285576, 2668289536

Offset: 1

Peter Bala, Sep 13 2007

Compare with A000292 and A040977 for the corresponding sums for the Vandermonde matrices of orders 2 and 3 respectively.

a(n) = sum of dimensions of all irreducible polynomial representations of GL(4) whose highest weight is of the form (m1 >= m2 >= m3 >= m4) and m1 <= n. - Oded Yacobi (oyacobi(AT)math.ucsd.edu), Jul 24 2008

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A000292, A040977, A107396, A133112.

A133111:= func< n | Binomial(n+2,5)*Binomial(n+4,5)/21 >;
[A133111(n): n in [0..30]]; // G. C. Greubel, Feb 09 2025

a[n_] := n^2 (n^2 - 1)^2 (n^2 - 4) (n^2 - 9)/302400; Array[a, 30] (* Robert G. Wilson v, Sep 17 2007 *)
Rest@ CoefficientList[ Series[x^4*(1 + 5 x + 5 x^2 + x^3)/(1 - x)^11, {x, 0, 30}], x] (* Robert G. Wilson v, Sep 17 2007 *)

def A133111(n): return binomial(n+2,5)*binomial(n+4,5)//21
print([A133111(n) for n in range(31)]) # G. C. Greubel, Feb 09 2025

a(n) = (1/288)*Sum_{1 <= i,j,k,l <= n} |(i-j)(i-k)(j-k)(i-l)(j-l)(k-l)|.
G.f.: x^4*(1 + 5*x + 5*x^2 + x^3)/(1 - x)^11 .
a(n) = n^2*(n^2 - 1)^2*(n^2 - 4)*(n^2 - 9)/302400.
a(n) = Sum_{i + j + k + l = n} i*j*k^2*l^3.
a(n) = (1/21)*A107396(n-3). - G. C. Greubel, Feb 09 2025

More terms from Robert G. Wilson v, Sep 17 2007

A124051 Quasi-mirror of A062196 formatted as a triangular array.

Original entry on oeis.org

3, 6, 8, 10, 30, 15, 15, 80, 90, 24, 21, 175, 350, 210, 35, 28, 336, 1050, 1120, 420, 48, 36, 588, 2646, 4410, 2940, 756, 63, 45, 960, 5880, 14112, 14700, 6720, 1260, 80, 55, 1485, 11880, 38808, 58212, 41580, 13860, 1980, 99, 66, 2200, 22275, 95040, 194040, 199584, 103950, 26400, 2970, 120

Offset: 0

Zerinvary Lajos, Nov 03 2006

			Triangle begins as:
   3;
   6,    8;
  10,   30,    15;
  15,   80,    90,    24;
  21,  175,   350,   210,     35;
  28,  336,  1050,  1120,    420,     48;
  36,  588,  2646,  4410,   2940,    756,     63;
  45,  960,  5880, 14112,  14700,   6720,   1260,    80;
  55, 1485, 11880, 38808,  58212,  41580,  13860,  1980,   99;
  66, 2200, 22275, 95040, 194040, 199584, 103950, 26400, 2970, 120;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000108, A000894, A062196, A286033.
Columns k: A000217(n+2) (k=0), A002417(n+1) (k=1), A001297(n) (k=2), A105946(n-2) (k=3), A105947(n-3) (k=4), A105948(n-4) (k=5), A107319(n-5) (k=6).
Diagonals: A005563(n+1) (k=n), A033487(n) (k=n-1), A027790(n) (k=n-2), A107395(n-3) (k=n-3), A107396(n-4) (k=n-4), A107397(n-5) (k=n-5), A107398(n-6) (k=n-6), A107399(n-7) (k=n-7).
Sums: A322938(n+1) (row).

A124051:= func< n,k | Binomial(n+1,n-k+1)*Binomial(n+3,n-k+1) >;
[A124051(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2025

for n from 0 to 10 do seq(binomial(n,i-1)*binomial(n+2,n+1-i), i=1..n ) od;

A124051[n_, k_]:= Binomial[n+1,n-k+1]*Binomial[n+3,n-k+1];
Table[A124051[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 07 2025 *)

def A124051(n,k): return binomial(n+1,n-k+1)*binomial(n+3,n-k+1)
print(flatten([[A124051(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 07 2025

From G. C. Greubel, Feb 07 2025: (Start)
T(n, k) = binomial(n+1, n-k+1)*binomial(n+3, n-k+1).
T(2*n, n) = (1/2)*A000894(n) + (5/2)*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*( (1+(-1)^n)*(-1)^(n/2)*A286033((n+4)/2) + (1-(-1)^n)*((-1)^((n+1)/2)*A000108((n+1)/2) - 1) ). (End)

cp's OEIS Frontend

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

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A062196 Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).

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A133111 a(n) = (1/(1!*2!*3!*4!))*Sum_{1 <= x_1, x_2, x_3, x_4 <= n} |det V(x_1,x_2,x_3,x_4)|, where V(x_1,x_2,x_3,x_4) is the Vandermonde matrix of order 4.

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A124051 Quasi-mirror of A062196 formatted as a triangular array.

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

A133111 a(n) = (1/(1!2!3!4!))Sum_{1 <= x_1, x_2, x_3, x_4 <= n} |det V(x_1,x_2,x_3,x_4)|, where V(x_1,x_2,x_3,x_4) is the Vandermonde matrix of order 4.