A001249 -id:A001249 - cp's OEIS frontend

A099764 a(n) = n^2 * (n+1)^2 * (n+2)^2 = 36*A001249(n-1).

0, 36, 576, 3600, 14400, 44100, 112896, 254016, 518400, 980100, 1742400, 2944656, 4769856, 7452900, 11289600, 16646400, 23970816, 33802596, 46785600, 63680400, 85377600, 112911876, 147476736, 190440000, 243360000, 308002500, 386358336

Offset: 0

Kari Lajunen (Kari.Lajunen(AT)Welho.com), Nov 11 2004

			a(0) = 1^3 - 1^3 = 0;
a(1) = (1+3)^3 - (1^3+3^3) = 64 - 28 = 36;
a(2) = (1+3+5)^3 - (1^3+3^3+5^3) = 729 - 153 = 576;
a(3) = (1+3+5+7)^3 - (1^3+3^3+5^3+7^3) = 4096 - 496 = 3600;
a(4) = (1+3+5+7+9)^3 - (1^3+3^3+5^3+7^3+9^3) = 15625 - 1225 = 14400; etc. - _Philippe Deléham_, Mar 10 2014

Jolley, Summation of Series, Dover (1961).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A001014, A002593.

List([1..30], n-> (n*(n^2-1))^2); # G. C. Greubel, Sep 03 2019

[n^2*(n+1)^2*(n+2)^2: n in [0..30]]; // Vincenzo Librandi, Oct 04 2011

A099764:=n->n^2*(n+1)^2*(n+2)^2; seq(A099764(n), n=0..30); # Wesley Ivan Hurt, Mar 09 2014

Table[n^2*(n+1)^2*(n+2)^2, {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
Times@@@Partition[Range[0,30]^2,3,1] (* Harvey P. Dale, Sep 02 2016 *)

vector(30, n, (n*(n^2-1))^2) \\ G. C. Greubel, Sep 03 2019

[(n*(n^2-1))^2 for n in (1..30)] # G. C. Greubel, Sep 03 2019

Sum_{n>=1} 1/a(n) = Pi^2/4-39/16 =  0.029901100272... [Jolley eq 241]
G.f.: 36*x*(1+x)*(1 +8*x +x^2)/(1-x)^7 . - R. J. Mathar, Oct 03 2011
a(n) = (Sum_{k=0..n} (2*k+1))^3 - Sum_{k=0..n} (2*k+1)^3. - Philippe Deléham, Mar 10 2014
a(n) = A001014(n+1) - A002593(n+1). - Philippe Deléham, Mar 10 2014
E.g.f.: exp(x)*x*(36+252*x+330*x^2+138*x^3+21*x^4+x^5). - Stefano Spezia, Sep 04 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/24 - 7/16. - Amiram Eldar, Jul 02 2020

A008459 Square the entries of Pascal's triangle.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 36, 16, 1, 1, 25, 100, 100, 25, 1, 1, 36, 225, 400, 225, 36, 1, 1, 49, 441, 1225, 1225, 441, 49, 1, 1, 64, 784, 3136, 4900, 3136, 784, 64, 1, 1, 81, 1296, 7056, 15876, 15876, 7056, 1296, 81, 1, 1, 100, 2025, 14400, 44100, 63504, 44100, 14400, 2025, 100, 1

Offset: 0

N. J. A. Sloane

Number of lattice paths from (0, 0) to (n, n) with steps (1, 0) and (0, 1), having k right turns. - Emeric Deutsch, Nov 23 2003
Product of A007318 and A105868. - Paul Barry, Nov 15 2005
Number of partitions that fit in an n X n box with Durfee square k. - Franklin T. Adams-Watters, Feb 20 2006
From Peter Bala, Oct 23 2008: (Start)
Narayana numbers of type B. Row n of this triangle is the h-vector of the simplicial complex dual to an associahedron of type B_n (a cyclohedron) [Fomin & Reading, p. 60]. See A063007 for the corresponding f-vectors for associahedra of type B_n. See A001263 for the h-vectors for associahedra of type A_n. The Hilbert transform of this triangular array is A108625 (see A145905 for the definition of this term).
Let A_n be the root lattice generated as a monoid by {e_i - e_j: 0 <= i, j <= n + 1}. Let P(A_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the h-vectors of a unimodular triangulation of P(A_n) [Ardila et al.]. A063007 is the corresponding array of f-vectors for these type A_n polytopes. See A086645 for the array of h-vectors for type C_n polytopes and A108558 for the array of h-vectors associated with type D_n polytopes.
(End)
The n-th row consists of the coefficients of the polynomial P_n(t) = Integral_{s = 0..2*Pi} (1 + t^2 - 2*t*cos(s))^n/Pi/2 ds. For example, when n = 3, we get P_3(t) = t^6 + 9*t^4 + 9*t^2 + 1; the coefficients are 1, 9, 9, 1. - Theodore Kolokolnikov, Oct 26 2010
Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0, 2*sqrt(-y)). Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence n!^2 as defined in Wang and Wang. - Peter Bala, Jul 24 2013
From Colin Defant, Sep 16 2018: (Start)
Let s denote West's stack-sorting map. T(n,k) is the number of permutations pi of [n+1] with k descents such that s(pi) avoids the patterns 132, 231, and 321. T(n,k) is also the number of permutations pi of [n+1] with k descents such that s(pi) avoids the patterns 132, 312, and 321.
T(n,k) is the number of permutations of [n+1] with k descents that avoid the patterns 1342, 3142, 3412, and 3421. (End)
The number of convex polyominoes whose smallest bounding rectangle has size (k+1)*(n+1-k) and which contain the lower left corner of the bounding rectangle (directed convex polyominoes). - Günter Rote, Feb 27 2019
Let P be the poset [n] X [n] ordered by the product order.  T(n,k) is the number of antichains in P containing exactly k elements. Cf. A063746. - Geoffrey Critzer, Mar 28 2020

			Pascal's triangle begins
  1
  1  1
  1  2   1
  1  3   3   1
  1  4   6   4   1
  1  5  10  10   5   1
  1  6  15  20  15   6   1
  1  7  21  35  35  21   7   1
...
so the present triangle begins
  1
  1   1
  1   4    1
  1   9    9     1
  1  16   36    16     1
  1  25  100   100    25    1
  1  36  225   400   225   36   1
  1  49  441  1225  1225  441  49   1
...

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 12.
J. Riordan, An introduction to combinatorial analysis, Dover Publications, Mineola, NY, 2002, page 191, Problem 15. MR1949650
P. G. Tait, On the Linear Differential Equation of the Second Order, Proceedings of the Royal Society of Edinburgh, 9 (1876), 93-98 (see p. 97) [From Tom Copeland, Sep 09 2010, vol number corrected Sep 10 2010]

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
N. Alexeev and A. Tikhomirov, Singular Values Distribution of Squares of Elliptic Random Matrices and type-B Narayana Polynomials, arXiv preprint arXiv:1501.04615 [math.PR], 2015.
F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices, arXiv:0809.5123 [math.CO], 2008.
Peter Bala, A commutative diagram of triangular arrays
E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005), 62-78.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
Carl M. Bender and Gerald V. Dunne, Polynomials and operator orderings, J. Math. Phys. 29 (1988), 1727-1731.
Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, and André Schulz, The Number of Convex Polyominoes with Given Height and Width, arXiv:1903.01095 [math.CO], 2019.
John H. Conway and N. J. A. Sloane, Low-dimensional lattices. VII Coordination sequences, Proc. R. Soc. Lond. A (1997) 453, 2369-2389.
R. Cori and G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
R. Cori and G. Hetyei, How to count genus one partitions, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Sergey Fomin and Nathan Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005, 2008. [From _Peter Bala_, Oct 23 2008]
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 10, 13.
Abdelkader Necer, Séries formelles et produit de Hadamard, Journal de théorie des nombres de Bordeaux, 9:2 (1997), pp. 319-335.
Weiping Wang and Tianming Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
Yi Wang and Arthur L.B. Yang, Total positivity of Narayana matrices, arXiv:1702.07822 [math.CO], 2017.
Harold R. L. Yang and Philip B. Zhang, Stable multivariate Narayana polynomials and labeled plane trees, arXiv:2403.15058 [math.CO], 2024. See p. 2.

Row sums are in A000984. Columns 0-3 are A000012, A000290, A000537, A001249.
Family of polynomials (see A062145): this sequence (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
Cf. A007318, A055133, A116647, A001263, A086645, A063007, A108558, A108625 (Hilbert transform), A145903, A181543, A086645 (logarithmic derivative), A105868 (inverse binomial transform), A093118.

Flat(List([0..10],n->List([0..n],k->Binomial(n,k)^2))); # Muniru A Asiru, Mar 30 2018

/* As triangle */ [[Binomial(n, k)^2: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Dec 15 2016

seq(seq(binomial(n, k)^2, k=0..n), n=0..10);

Table[Binomial[n, k]^2, {n, 0, 11}, {k, 0, n}]//Flatten (* Alonso del Arte, Dec 08 2013 *)

create_list(binomial(n,k)^2,n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

T(n,k):=if n=k then 1 else if k=0 then 1 else T(n-1,k)*(n+k)/(n-k)+T(n-1,k-1); /* Vladimir Kruchinin, Oct 18 2014 */

A(x,y):=1/sqrt(1-2*x-2*x*y+x^2-2*x^2*y+x^2*y^2);
taylor(x*A(x,y)+x*y*A(x,y)+sqrt(1+4*x^2*y*A(x,y)^2),x,0,7,y,0,7); /* Vladimir Kruchinin, Oct 23 2020 */

{T(n, k) = if( k<0 || k>n, 0, binomial(n, k)^2)}; /* Michael Somos, May 03 2004 */

{T(n,k)=polcoeff(polcoeff(sum(m=0,n,(2*m)!/m!^2*x^(2*m)*y^m/(1-x-x*y+x*O(x^n))^(2*m+1)),n,x),k,y)} \\ Paul D. Hanna, Oct 31 2010

def A008459(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),n-comb(r+1,2))**2 # Chai Wah Wu, Nov 12 2024

T(n,k) = A007318(n,k)^2. - Sean A. Irvine, Mar 29 2018
E.g.f.: exp((1+y)*x)*BesselI(0, 2*sqrt(y)*x). - Vladeta Jovovic, Nov 17 2003
G.f.: 1/sqrt(1-2*x-2*x*y+x^2-2*x^2*y+x^2*y^2); g.f. for row n: (1-t)^n P_n[(1+t)/(1-t)] where the P_n's are the Legendre polynomials. - Emeric Deutsch, Nov 23 2003 [The original version of the bivariate g.f. has been modified with the roles of x and y interchanged so that now x corresponds to n and y to k. - Petros Hadjicostas, Oct 22 2017]
G.f. for column k is Sum_{j = 0..k} C(k, j)^2*x^(k+j)/(1 - x)^(2*k+1). - Paul Barry, Nov 15 2005
Column k has g.f. (x^k)*Legendre_P(k, (1+x)/(1-x))/(1 - x)^(k+1) = (x^k)*Sum_{j = 0..k} C(k, j)^2*x^j/(1 - x)^(2*k+1). - Paul Barry, Nov 19 2005
Let E be the operator D*x*D, where D denotes the derivative operator d/dx. Then (1/n!^2) * E^n(1/(1 - x)) = (row n generating polynomial)/(1 - x)^(2*n+1) = Sum_{k >= 0} binomial(n+k, k)^2*x^k. For example, when n = 3 we have (1/3!)^2*E^3(1/(1 - x)) = (1 + 9*x + 9*x^2 + x^3)/(1 - x)^7 = (1/3!)^2 * Sum_{k >= 0} ((k+1)*(k+2)*(k+3))^2*x^k. - Peter Bala, Oct 23 2008
G.f.: A(x, y) = Sum_{n >= 0} (2*n)!/n!^2 * x^(2*n)*y^n/(1 - x - x*y)^(2*n+1). - Paul D. Hanna, Oct 31 2010
From Peter Bala, Jul 24 2013: (Start)
Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0, 2*sqrt(-y)). Generating function: E(y)*E(x*y) = 1 + (1 + x)*y + (1 + 4*x + x^2)*y^2/2!^2 + (1 + 9*x + 9*x^2 + x^3)*y^3/3!^2 + .... Cf. the unsigned version of A021009 with generating function exp(y)*E(x*y).
The n-th power of this array has the generating function E(y)^n*E(x*y). In particular, the matrix inverse A055133 has the generating function E(x*y)/E(y). (End)
T(n,k) = T(n-1,k)*(n+k)/(n-k) + T(n-1,k-1), T(n,0) = T(n,n) = 1. - Vladimir Kruchinin, Oct 18 2014
Observe that the recurrence T(n,k) = T(n-1,k)*(n+k)/(n-k) - T(n-1,k-1), for n >= 2 and 1 <= k < n, with boundary conditions T(n,0) = T(n,n) = 1 gives Pascal's triangle A007318. - Peter Bala, Dec 21 2014
n-th row polynomial R(n, x) = [z^n] (1 + (1 + x)*z + x*z^2)^n. Note that 1/n*[z^(n-1)] (1 + (1 + x)*z + x*z^2)^n gives the row polynomials of A001263. - Peter Bala, Jun 24 2015
Binomial transform of A105868. If G(x,t) = 1/sqrt(1 - 2*(1 + t)*x + (1 - t)^2*x^2) denotes the o.g.f. of this array then 1 + x*d/dx log(G(x,t)) = 1 + (1 + t)*x + (1 + 6*t + t^2)*x^2 + ... is the o.g.f. for A086645. - Peter Bala, Sep 06 2015
T(n,k) = Sum_{i=0..n} C(n-i,k)*C(n,i)*C(n+i,i)*(-1)^(n-i-k). - Vladimir Kruchinin, Jan 14 2018
G.f. satisfies A(x,y) = x*A(x,y)+x*y*A(x,y)+sqrt(1+4*x^2*y*A(x,y)^2). - Vladimir Kruchinin, Oct 23 2020
G.f. satisfies the differential equation y * d^2(A(x,y))/dy^2 - x^2 * d^2(x*A(x,y))/dx^2 + 2*x^2* A(x,y)^3 = 0. - Sergii Voloshyn, Mar 07 2025
T(n,k) = Sum_{i=0..n} C(2*n+1,i)*C(n+k-i,n)^2*(-1)^i. - Natalia L. Skirrow, Apr 14 2025

A086020 a(n) = Sum_(i=1..n) binomial(i+2,3)^2 [ Sequential sums of the tetragonal numbers or "tetras" (pyramidal, square) raised to power 2 (drawn from the 4th diagonal - left or right - of Pascal's Triangle) ].

Original entry on oeis.org

1, 17, 117, 517, 1742, 4878, 11934, 26334, 53559, 101959, 183755, 316251, 523276, 836876, 1299276, 1965132, 2904093, 4203693, 5972593, 8344193, 11480634, 15577210, 20867210, 27627210, 36182835, 46915011, 60266727, 76750327

Offset: 1

André F. Labossière, Jul 17 2003

Kekulé numbers for certain benzenoids (see the Cyvin-Gutman reference, p. 243; expression in (13.26) yields same sequence with offset 0). - Emeric Deutsch, Aug 02 2005

Partial sums of A001249. - R. J. Mathar, Aug 19 2008

			a(8) = Sum_{i=1..8} binomial(i+2,3)^2 = (20*(8^7) + 210*(8^6) + 854*(8^5) + 1680*(8^4) + 1610*(8^3) + 630*(8^2) + 36*8)/7! = 26334.

T. D. Noe, Table of n, a(n) for n = 1..1000
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988.
John Engbers and Christopher Stocker, Two Combinatorial Proofs of Identities Involving Sums of Powers of Binomial Coefficients, Integers 16 (2016), #A58.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000292, A087127, A024166, A085438, A085439, A085440, A085441, A085442, A000332, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030.

[n*(n+1)*(n+2)*(n+3)*(2*n+3)*(5*n^2+15*n+1)/2520: n in [1..30]]; // G. C. Greubel, Nov 22 2017

a:=n->n*(n+1)*(n+2)*(n+3)*(2*n+3)*(5*n^2+15*n+1)/2520: seq(a(n),n=1..31); # Emeric Deutsch

Accumulate[Binomial[Range[30]+2,3]^2]  (* Harvey P. Dale, Mar 24 2011 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,17,117,517,1742,4878, 11934, 26334},30] (* Harvey P. Dale, Aug 17 2014 *)

a(n)=n*(n+1)*(n+2)*(n+3)*(2*n+3)*(5*n^2+15*n+1)/2520 \\ Charles R Greathouse IV, May 18 2015

a(n) = Sum_(i=1..n) binomial(i+2, 3)^2.
a(n) = ( C(n+3, 4)/35 )*( 35 + 84*C(n-1, 1) + 70*C(n-1, 2) + 20*C(n-1, 3) ).
a(n) = n*(n+1)*(n+2)*(n+3)*(2*n+3)(5*n^2 + 15*n + 1)/2520. - Emeric Deutsch, Aug 02 2005
O.g.f: x*(1+x)*(1 + 8*x + x^2)/(1-x)^8. - R. J. Mathar, Aug 19 2008

A033455 Convolution of nonzero squares A000290 with themselves.

Original entry on oeis.org

1, 8, 34, 104, 259, 560, 1092, 1968, 3333, 5368, 8294, 12376, 17927, 25312, 34952, 47328, 62985, 82536, 106666, 136136, 171787, 214544, 265420, 325520, 396045, 478296, 573678, 683704, 809999, 954304, 1118480, 1304512, 1514513, 1750728, 2015538, 2311464

Offset: 1

N. J. A. Sloane

Total area of all square regions from an n X n grid. E.g., at n = 3, there are nine individual squares, four 2 X 2's and one 3 X 3, total area 9 + 16 + 9 = 34, hence a(3) = 34. - Jon Perry, Jul 29 2003
If X is an n-set and Y and Z disjoint 2-subsets of X then a(n) is equal to the number of 7-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
Every fourth term is odd. However, there are no primes in the sequence. - Zak Seidov, Feb 28 2011
-120*a(n) is the real part of (n + n*i)*(n + 2 + n*i)*(n + (n + 2)i)*(n + 2+(n + 2)*i)*(n + 1 + (n + 1)*i), where i = sqrt(-1). - Jon Perry, Feb 05 2014
The previous formula rephrases the factorization of the 5th-order polynomial a(n) = (n+1)*((n+1)^4-1) = (n+1)*A123864(n+1) based on the factorization in A123865. - R. J. Mathar, Feb 08 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Abderrahim Arabi, Hacène Belbachir, Jean-Philippe Dubernard, Plateau Polycubes and Lateral Area, arXiv:1811.05707 [math.CO], 2018. See Column 1 Table 1 p. 8.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 2. - _N. J. A. Sloane_, Mar 23 2014
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000290, A001249, A123865, A219086.

List([1..40], n-> ((n+1)^5 -(n+1))/30); # G. C. Greubel, Jul 05 2019

[(n^5-n)/30: n in [2..41]]; // Vincenzo Librandi, Mar 24 2014

Table[(n^5 - n)/30, {n,2,41}] (* Vladimir Joseph Stephan Orlovsky, Feb 28 2011 *)
CoefficientList[Series[(1+x)^2/(1-x)^6, {x,0,40}], x] (* Vincenzo Librandi, Mar 24 2014 *)

vector(40, n, ((n+1)^5 -n-1)/30) \\ G. C. Greubel, Jul 05 2019

[((n+1)^5 -n-1)/30 for n in (1..40)] # G. C. Greubel, Jul 05 2019

a(n-1) = n*(n^4 - 1)/30 = A061167(n)/30. - Henry Bottomley, Apr 18 2001
G.f.: x*(1+x)^2/(1-x)^6. - Philippe Deléham, Feb 21 2012
a(n) = Sum_{k=1..n+1} k^2*(n+1-k)^2. - Kolosov Petro, Feb 07 2019
E.g.f.: x*(30 +90*x +65*x^2 +15*x^3 +x^4)*exp(x)/30. - G. C. Greubel, Jul 05 2019

More terms from Vincenzo Librandi, Mar 24 2014

A250853 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Original entry on oeis.org

100, 400, 543, 1225, 2457, 2670, 3136, 8037, 13097, 12311, 7056, 21436, 44797, 63631, 54410, 14400, 49599, 123016, 223933, 291165, 233683, 27225, 103293, 290646, 626416, 1043885, 1280447, 983950, 48400, 198297, 614965, 1499679, 2955136

Offset: 1

R. H. Hardin, Nov 28 2014

Table starts
......100.......400.......1225.......3136........7056.......14400.......27225
......543......2457.......8037......21436.......49599......103293......198297
.....2670.....13097......44797.....123016......290646......614965.....1195457
....12311.....63631.....223933.....626416.....1499679.....3204951.....6279401
....54410....291165....1043885....2955136.....7134786....15344785....30214465
...233683...1280447....4648157...13263136....32201019....69543783...137379337
...983950...5480917...20067117...57570016...140301126...303858745...601566177
..4085631..23024631...84805533..244213216...596722599..1294875471..2567402601
.16796370..95448605..353060845.1019415136..2495502666..5422612945.10763029505
.68555723.391939087.1454214877.4206874336.10311967539.22429374423.44552408777

			Some solutions for n=3 k=4
..2..2..0..0..0....1..2..3..2..2....2..2..1..0..0....3..2..1..1..1
..0..0..0..0..0....0..1..2..2..3....0..0..0..0..0....0..0..0..0..0
..1..1..1..1..1....0..1..2..2..3....1..1..1..1..3....0..0..1..1..2
..0..1..1..1..3....0..1..2..2..3....0..0..0..1..3....0..0..1..1..2

R. H. Hardin, Table of n, a(n) for n = 1..160

Row 1 is A001249(n+1)

Empirical T(n,k) = (((31/36)*k^6+(25/2)*k^5+(1229/18)*k^4+(620/3)*k^3+(10759/36)*k^2+(1181/6)*k+48)*4^n -((5/3)*k^6+(133/6)*k^5+(320/3)*k^4+(1717/6)*k^3+(944/3)*k^2+(344/3)*k)*3^n +(k^6+12*k^5+47*k^4+103*k^3+54*k^2-13*k)*2^n -((1/9)*k^6+(3/2)*k^5+(25/9)*k^4+(13/6)*k^3-(89/9)*k^2+(4/3)*k))/12
Empirical for column k:
k=1: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (832*4^n-846*3^n+204*2^n+2)/12
k=2: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (4838*4^n-6300*3^n+2214*2^n-80)/12
k=3: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (18104*4^n-26144*3^n+10680*2^n-644)/12
k=4: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (52650*4^n-80640*3^n+35820*2^n-2688)/12
k=5: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (129528*4^n-206190*3^n+96660*2^n-8190)/12
k=6: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (282492*4^n-462196*3^n+224994*2^n-20568)/12
k=7: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (562288*4^n-939120*3^n+470064*2^n-45220)/12
Empirical for row n:
n=1: a(n) = (1/36)*n^6 + (1/2)*n^5 + (133/36)*n^4 + (43/3)*n^3 + (277/9)*n^2 + (104/3)*n + 16
n=2: a(n) = (2/9)*n^6 + (47/12)*n^5 + (953/36)*n^4 + (1141/12)*n^3 + (6527/36)*n^2 + 172*n + 64
n=3: a(n) = (3/2)*n^6 + (74/3)*n^5 + (621/4)*n^4 + (3161/6)*n^3 + (3691/4)*n^2 + 783*n + 256
n=4: a(n) = (76/9)*n^6 + (1595/12)*n^5 + (28765/36)*n^4 + (31373/12)*n^3 + (155683/36)*n^2 + (10223/3)*n + 1024
n=5: a(n) = (763/18)*n^6 + (1949/3)*n^5 + (136493/36)*n^4 + (72691/6)*n^3 + (693923/36)*n^2 + (43319/3)*n + 4096
n=6: a(n) = 198*n^6 + (35807/12)*n^5 + (204911/12)*n^4 + (214827/4)*n^3 + (998209/12)*n^2 + (180451/3)*n + 16384
n=7: a(n) = (15887/18)*n^6 + (39464/3)*n^5 + (2674189/36)*n^4 + (462227/2)*n^3 + (12645859/36)*n^2 + (743119/3)*n + 65536

A223864 T(n,k)=Number of nXk 0..3 arrays with rows and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

4, 16, 10, 50, 100, 20, 130, 684, 400, 35, 296, 3526, 4884, 1225, 56, 610, 14751, 41682, 24199, 3136, 84, 1163, 52591, 273959, 315124, 93731, 7056, 120, 2083, 165212, 1477240, 3017129, 1771012, 303560, 14400, 165, 3544, 468292, 6818350, 22852913

Offset: 1

R. H. Hardin Mar 28 2013

Table starts
...4....16.......50.......130.........296...........610...........1163
..10...100......684......3526.......14751.........52591.........165212
..20...400.....4884.....41682......273959.......1477240........6818350
..35..1225....24199....315124.....3017129......22852913......144081276
..56..3136....93731...1771012....23738426.....243933798.....2030417942
..84..7056...303560...8008548...145947740....1989679315....21476594002
.120.14400...857696..30627033...740441932...13140481520...181330154458
.165.27225..2175884.102479569..3217594840...73068868012..1271807435844
.220.48400..5058530.307435001.12305144319..352040804450..7630189031428
.286.81796.10940664.842078930.42270004211.1502130487437.40055722078772

			Some solutions for n=3 k=4
..0..0..0..0....0..2..1..1....0..1..3..2....0..2..2..0....0..1..1..3
..0..0..0..1....0..2..3..1....0..2..3..2....1..3..3..2....0..1..3..3
..2..3..2..2....0..2..3..3....1..3..3..3....3..3..3..2....1..3..3..3

R. H. Hardin, Table of n, a(n) for n = 1..312

Column 1 is A000292(n+1)
Column 2 is A001249
Row 1 is A223659

Empirical: columns k=1..7 are polynomials of degree 3*k for n>0,0,0,1,2,4,6

Empirical: rows n=1..7 are polynomials of degree 6*n

A224173 T(n,k) = number of n X k 0..3 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

4, 16, 10, 50, 100, 20, 130, 684, 400, 35, 296, 3526, 4739, 1225, 56, 610, 14751, 38561, 22988, 3136, 84, 1163, 52591, 242114, 272130, 87878, 7056, 120, 2083, 165212, 1253770, 2335459, 1460836, 282372, 14400, 165, 3544, 468292, 5588411, 15925611

Offset: 1

R. H. Hardin, Mar 31 2013

			Table starts:
    4    16       50       130         296          610           1163
   10   100      684      3526       14751        52591         165212
   20   400     4739     38561      242114      1253770        5588411
   35  1225    22988    272130     2335459     15925611       91494280
   56  3136    87878   1460836    16625026    143558572     1012166273
   84  7056   282372   6425876    95808564   1038484760     8857798353
  120 14400   794220  24197608   468021427   6360047093    65713691148
  165 27225  2010035  80350989  1994287334  33901838632   426013124302
  220 48400  4668304 240416852  7568051210 160168789130  2451904991177
  286 81796 10095924 658890738 25994968917 680269560125 12667946702827
  ...
Some solutions for n=3 k=4
..0..0..1..0....0..0..1..2....0..0..3..0....0..2..0..0....0..3..3..1
..1..3..3..1....0..1..3..2....3..3..3..1....1..2..0..0....1..3..3..1
..1..3..3..3....0..3..3..2....3..3..3..2....2..2..1..0....1..3..3..3

R. H. Hardin, Table of n, a(n) for n = 1..178

Main diagonal is A224167.
Columns 1..7 are A000292(n+1), A001249, A224168, A224169, A224170, A224171, A224172.
Rows 1..7 are A223659, A223865, A224174, A224175, A224176, A224177, A224178.
Cf. A223838.

Empirical: columns k=1..7 are polynomials of degree 3*k for n>0,0,0,3,6,9,12.

Empirical: rows n=1..5 are polynomials of degree 6*n for k>0,0,0,2,6.

Name corrected by Andrew Howroyd, Mar 18 2025

A223987 T(n,k)=Number of nXk 0..3 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

4, 16, 10, 50, 100, 20, 130, 684, 400, 35, 296, 3526, 5029, 1225, 56, 610, 14751, 44803, 25410, 3136, 84, 1163, 52591, 308470, 358118, 99634, 7056, 120, 2083, 165212, 1738756, 3770722, 2086196, 325120, 14400, 165, 3544, 468292, 8350154, 31585056

Offset: 1

R. H. Hardin Mar 30 2013

Table starts
...4....16.......50........130.........296...........610...........1163
..10...100......684.......3526.......14751.........52591.........165212
..20...400.....5029......44803......308470.......1738756........8350154
..35..1225....25410.....358118.....3770722......31585056......219861244
..56..3136....99634....2086196....31831914.....378122264.....3661410444
..84..7056...325120....9647292...204647416....3322756326....43307637038
.120.14400...922768...37395816..1067023886...22985966340...392525216516
.165.27225..2346883..126087157..4710529013..131366850521..2873859236297
.220.48400..5462600..379654704.18159308422..642224541548.17659521902693
.286.81796.11818092.1040942916.62548820489.2756467192963.93729371629362

			Some solutions for n=3 k=4
..0..2..1..1....0..0..2..0....0..1..1..0....1..2..0..0....2..2..2..0
..0..2..2..1....1..1..2..0....0..3..2..1....1..2..2..0....3..2..2..1
..1..3..3..3....2..3..2..1....3..3..2..1....2..3..3..2....3..2..2..1

R. H. Hardin, Table of n, a(n) for n = 1..420

Column 1 is A000292(n+1)
Column 2 is A001249
Row 1 is A223659
Row 2 is A223865

Empirical: columns k=1..7 are polynomials of degree 3*k

Empirical: rows n=1..7 are polynomials of degree 6*n

A224024 T(n,k)=Number of nXk 0..3 arrays with rows nondecreasing and antidiagonals unimodal.

Original entry on oeis.org

4, 10, 16, 20, 100, 64, 35, 400, 1000, 256, 56, 1225, 6796, 10000, 1024, 84, 3136, 32523, 112436, 100000, 4096, 120, 7056, 122523, 772683, 1859020, 1000000, 16384, 165, 14400, 387729, 4002738, 17735200, 30756756, 10000000, 65536, 220, 27225, 1074167

Offset: 1

R. H. Hardin Mar 30 2013

Table starts
.......4..........10............20..............35...............56
......16.........100...........400............1225.............3136
......64........1000..........6796...........32523...........122523
.....256.......10000........112436..........772683..........4002738
....1024......100000.......1859020........17735200........120352359
....4096.....1000000......30756756.......403836633.......3491241557
...16384....10000000.....508916456......9186127249......99853876444
...65536...100000000....8420768936....208983591829....2841637297963
..262144..1000000000..139333478144...4754911670136...80738139650660
.1048576.10000000000.2305467501680.108190494364824.2292943314015674

			Some solutions for n=3 k=4
..3..3..3..3....1..3..3..3....0..0..0..2....1..1..2..2....0..0..1..1
..0..2..3..3....0..2..3..3....2..2..3..3....0..0..1..2....0..1..3..3
..1..1..1..1....0..0..1..1....0..2..2..3....0..0..2..3....1..1..3..3

R. H. Hardin, Table of n, a(n) for n = 1..364

Column 1 is A000302
Column 2 is A011557
Row 1 is A000292(n+1)
Row 2 is A001249

Empirical: columns k=1..7 have recurrences of order 1,1,7,10,19,25,41

Empirical: rows n=1..7 are polynomials of degree 3*n for k>0,0,1,2,3,4,5

A224391 T(n,k)=Number of nXk 0..3 arrays with diagonals and antidiagonals unimodal and rows nondecreasing.

Original entry on oeis.org

4, 10, 16, 20, 100, 64, 35, 400, 1000, 256, 56, 1225, 6094, 10000, 1024, 84, 3136, 27790, 86701, 100000, 4096, 120, 7056, 102232, 497958, 1268572, 1000000, 16384, 165, 14400, 319769, 2332222, 8573507, 18794636, 10000000, 65536, 220, 27225, 881519

Offset: 1

R. H. Hardin Apr 05 2013

Table starts
.......4..........10...........20.............35..............56
......16.........100..........400...........1225............3136
......64........1000.........6094..........27790..........102232
.....256.......10000........86701.........497958.........2332222
....1024......100000......1268572........8573507........45648753
....4096.....1000000.....18794636......152271025.......879830242
...16384....10000000....279128617.....2780848289.....17642791909
...65536...100000000...4142692993....51325449985....365858453951
..262144..1000000000..61481903024...949582166068...7713944320142
.1048576.10000000000.912523782542.17572045403455.163629606236587

			Some solutions for n=3 k=4
..0..1..1..1....0..1..1..3....1..2..3..3....2..2..3..3....0..0..0..1
..2..2..2..2....0..1..3..3....0..2..2..2....0..3..3..3....0..2..2..3
..1..3..3..3....0..2..3..3....1..1..1..1....0..0..1..2....3..3..3..3

R. H. Hardin, Table of n, a(n) for n = 1..198

Column 1 is A000302
Column 2 is A011557
Row 1 is A000292(n+1)
Row 2 is A001249

Empirical for column k:
k=1: a(n) = 4*a(n-1)
k=2: a(n) = 10*a(n-1)
k=3: [order 15]
k=4: [order 47]
Empirical: rows n=1..6 are polynomials of degree 3*n for k>0,0,1,4,7,10

cp's OEIS Frontend

A099764 a(n) = n^2 * (n+1)^2 * (n+2)^2 = 36*A001249(n-1).

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A008459 Square the entries of Pascal's triangle.

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A086020 a(n) = Sum_(i=1..n) binomial(i+2,3)^2 [ Sequential sums of the tetragonal numbers or "tetras" (pyramidal, square) raised to power 2 (drawn from the 4th diagonal - left or right - of Pascal's Triangle) ].

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A033455 Convolution of nonzero squares A000290 with themselves.

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A250853 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

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A223864 T(n,k)=Number of nXk 0..3 arrays with rows and antidiagonals unimodal and columns nondecreasing.

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