A005585 -id:A005585 - cp's OEIS frontend

A213561 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = m*(m+1)/2, m=n-1+h, n>=1, h>=1, and ** = convolution.

1, 7, 3, 27, 18, 6, 77, 61, 34, 10, 182, 157, 109, 55, 15, 378, 342, 267, 171, 81, 21, 714, 665, 557, 407, 247, 112, 28, 1254, 1190, 1043, 827, 577, 337, 148, 36, 2079, 1998, 1806, 1512, 1152, 777, 441, 189, 45, 3289, 3189, 2946, 2562, 2072, 1532

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal:  A213562
Antidiagonal sums:  A213563
Row 1,  (1,4,9,...)**(1,3,6,...):  A005585
Row 2,  (1,4,9,...)**(3,6,10,...):  (2*k^5 +25*k^4 + 120*k^3 + 155*k^2 + 58*k)/120
Row 3,  (1,4,9,...)**(6,10,15,...):  (2*k^5 +35*k^4 + 60*k^3 + 325*k^2 + 118*k)/120
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....7.....27....77....182
3....18....61....157...342
6....34....109...267...557
10...55....171...407...827
15...81....247...577...1152
21...112...337...777...1532

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A213500.

b[n_] := n^2; c[n_] := n (n + 1)/2
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213561 *)
d = Table[t[n, n], {n, 1, 40}] (* A213562 *)
s1 = Table[s[n], {n, 1, 50}] (* A213563 *)

T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) - T(n,k-6).

G.f. for row n: f(x)/g(x), where f(x) = n*(n + 1) - (n^2 - n - 2)*x - (n^2 + n - 2)*x^2 + n*(n - 1)*x^3 and g(x) = 2*(1 - x)^6.

A067056 a(n) = (1)(2 + 3 + 4 + ... + n) + (1 + 2)(3 + 4 + 5 + ... + n) + (1 + 2 + 3)(4 + 5 + 6 + ... + n) + ... + (1 + 2 + 3 + ... + n-1)n.

Original entry on oeis.org

1, 2, 14, 54, 154, 364, 756, 1428, 2508, 4158, 6578, 10010, 14742, 21112, 29512, 40392, 54264, 71706, 93366, 119966, 152306, 191268, 237820, 293020, 358020, 434070, 522522, 624834, 742574, 877424, 1031184, 1205776, 1403248, 1625778, 1875678, 2155398, 2467530

Offset: 1

Amarnath Murthy, Jan 02 2002

			a(4) = (1)*(2+3+4) + (1+2)*(3+4) + (1+2+3)*(4) = 9 + 21 + 24 = 54.

Colin Barker, Table of n, a(n) for n = 1..1000
Bünyamin Şahin, Level Polynomials of Rooted Trees, 2023.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000217, A001105, A005585.

Join[{1},Table[Total[Total[#[[1]]Total[#[[2]]]]&/@Table[TakeDrop[ Range[ k],n],{n,k-1}]],{k,2,40}]] (* Requires Mathematica version 10 or later *)  (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,2,14,54,154,364,756},40] (* Harvey P. Dale, Jul 17 2020 *)

t(n) = n*(n+1)/2;
a(n) = if (n=1, 1, sum(k=1, n-1, t(k)*(t(n) - t(k)))); \\ Michel Marcus, Mar 06 2018

Vec(x*(1 - 4*x + 17*x^2 - 20*x^3 + 15*x^4 - 6*x^5 + x^6) / (1 - x)^6 + O(x^60)) \\ Colin Barker, Mar 06 2018

a(n) = Sum_{r=1..n-1} t(r)*(t(n) - t(r)), where t(r) is the r-th triangular number, n>1.
a(n) = n*(2*n^4 + 5*n^3 - 5*n - 2)/60 = (n-1)*n*(n+1)*(n+2)*(2*n+1)/60, n>1. - Ralf Stephan, Apr 30 2004
a(n) = 2*A005585(n-1), n>1. - R. J. Mathar, May 20 2013
a(n) = Sum_{i=1..n} A000217(i)*A001105(n-i) for n>1, a(1)=1. - Bruno Berselli, Mar 06 2018
From Colin Barker, Mar 06 2018: (Start)
G.f.: x*(1 - 4*x + 17*x^2 - 20*x^3 + 15*x^4 - 6*x^5 + x^6) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>7.
(End)

More terms from Jason Earls, Jan 11 2002

A207543 Triangle read by rows, expansion of (1+yx)/(1-2y*x+y(y-1)x^2).

Original entry on oeis.org

1, 0, 3, 0, 1, 5, 0, 0, 5, 7, 0, 0, 1, 14, 9, 0, 0, 0, 7, 30, 11, 0, 0, 0, 1, 27, 55, 13, 0, 0, 0, 0, 9, 77, 91, 15, 0, 0, 0, 0, 1, 44, 182, 140, 17, 0, 0, 0, 0, 0, 11, 156, 378, 204, 19, 0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21, 0

Offset: 0

Philippe Deléham, Feb 24 2012

Previous name was: "A scaled version of triangle A082985."

Triangle, read by rows, given by (0, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

			Triangle begins :
1
0, 3
0, 1, 5
0, 0, 5, 7
0, 0, 1, 14, 9
0, 0, 0, 7, 30, 11
0, 0, 0, 1, 27, 55, 13
0, 0, 0, 0, 9, 77, 91, 15
0, 0, 0, 0, 1, 44, 182, 140, 17
0, 0, 0, 0, 0, 11, 156, 378, 204, 19
0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21
0, 0, 0, 0, 0, 0, 13, 275, 1122, 1254, 385, 23

R. Zielinski, Induction and Analogy in a Problem of Finite Sums arXiv:1608.04006 [math.GM], 2016.

Cf. A082985 which is another version of this triangle.

Cf. Diagonals : A005408, A000330, A005585, A050486, A054333, A057788. Cf. A119900.

s := (1+y*x)/(1-2*y*x+y*(y-1)*x^2): t := series(s,x,12):
seq(print(seq(coeff(coeff(t,x,n),y,m),m=0..n)),n=0..11); # Peter Luschny, Aug 17 2016

T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 3.
G.f.: (1+y*x)/(1-2*y*x+y*(y-1)*x^2).
Sum_{i, i>=0} T(n+i,n) = A000204(2*n+1).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A078069(n), A000007(n), A003945(n), A111566(n) for x = -1, 0, 1, 2 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A090131(n), A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively.
From Peter Bala, Aug 17 2016: (Start)
Let S(k,n) = Sum_{i = 1..n} i^k. Calculations in Zielinski 2016 suggest the following identity holds involving the p-th row elements of this triangle:
Sum_{k = 0..p} T(p,k)*S(2*k,n) = 1/2*(2*n + 1)*(n*(n + 1))^p.
For example, for row 6 we find S(6,n) + 27*S(8,n) + 55*S(10,n) + 13*S(12,n) = 1/2*(2*n + 1)*(n*(n + 1))^6.
There appears to be a similar result for the odd power sums S(2*k + 1,n) involving A119900. (End)

New name using a formula of the author from Peter Luschny, Aug 17 2016

A104711 Triangle T(n,m) = sum_{k=m..n} A001263(k,m).

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 10, 7, 1, 5, 20, 27, 11, 1, 6, 35, 77, 61, 16, 1, 7, 56, 182, 236, 121, 22, 1, 8, 84, 378, 726, 611, 218, 29, 1, 9, 120, 714, 1902, 2375, 1394, 365, 37, 1, 10, 165, 1254, 4422, 7667, 6686, 2885, 577, 46, 1, 11, 220, 2079, 9372, 21527, 26090, 16745

Offset: 1

Gary W. Adamson, Mar 19 2005

This summation over columns of the Narayana triangle could also be defined as a multiplication

of the Narayana triangle from the left by the lower-left triangle represented by the all-1 sequence A000012.

			First few rows of the triangle are:
1;
2, 1;
3, 4, 1;
4, 10, 7, 1;
5, 20, 27, 11, 1;
6, 35, 77, 61, 16, 1;
...

Cf. A014138, A104710, A005585, A000124.

from sympy import binomial
def A001263(n,m):
    return binomial(n-1,m-1)*binomial(n,m-1)//m
def A104711(n,m):
    a = 0
    for k in range(m,n+1):
        a += A001263(k,m)
    return a
print([A104711(n,m) for n in range(20) for m in range(1,n+1)]) # R. J. Mathar, Oct 11 2009

Row sums: sum_{m=1..n} T(n,m) = A014138(n).

Extended by R. J. Mathar, Oct 11 2009

A129710 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 2, 5, 1, 2, 7, 4, 2, 9, 9, 1, 2, 11, 16, 5, 2, 13, 25, 14, 1, 2, 15, 36, 30, 6, 2, 17, 49, 55, 20, 1, 2, 19, 64, 91, 50, 7, 2, 21, 81, 140, 105, 27, 1, 2, 23, 100, 204, 196, 77, 8, 2, 25, 121, 285, 336, 182, 35, 1, 2, 27, 144, 385, 540, 378, 112, 9, 2, 29, 169, 506

Offset: 0

Emeric Deutsch, May 12 2007

Also number of Fibonacci binary words of length n and having k 10 subwords.
Row n has 1+floor(n/2) terms.
Row sums are the Fibonacci numbers (A000045).
T(n,0)=2 for n >= 1.
Sum_{k>=0} k*T(n,k) = A023610(n-2).
Triangle, with zeros omitted, given by (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 14 2012
Riordan array ((1+x)/(1-x), x^2/(1-x)), zeros omitted. - Philippe Deléham, Jan 14 2012

			T(5,2)=4 because we have 10101, 01101, 01010 and 01011.
Triangle starts:
  1;
  2;
  2, 1;
  2, 3;
  2, 5, 1;
  2, 7, 4;
  2, 9, 9, 1;
Triangle (2, -1, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) begins:
  1;
  2, 0;
  2, 1, 0;
  2, 3, 0, 0;
  2, 5, 1, 0, 0;
  2, 7, 4, 0, 0, 0;
  2, 9, 9, 1, 0, 0, 0;

Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 200, flattened.)
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Thomas Grubb and Frederick Rajasekaran, Set Partition Patterns and the Dimension Index, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence.

Cf. A000045, A023610.
Cf. A029635, A029653.
Columns: A040000, A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.

T:=proc(n,k) if n=0 and k=0 then 1 elif k<=floor(n/2) then binomial(n-k,k)+binomial(n-k-1,k) else 0 fi end: for n from 0 to 18 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

MapAt[# - 1 &, #, 1] &@ Table[Binomial[n - k, k] + Binomial[n - k - 1, k], {n, 0, 16}, {k, 0, Floor[n/2]}] // Flatten (* Michael De Vlieger, Nov 15 2019 *)

T(n,k) = binomial(n-k,k) + binomial(n-k-1,k) for n >= 1 and 0 <= k <= floor(n/2).
G.f. = G(t,z) = (1+z)/(1-z-tz^2).
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A078050(n), A057079(n), A040000(n), A000045(n+2), A000079(n), A006138(n), A026597(n), A133407(n), A133467(n), A133469(n), A133479(n), A133558(n), A133577(n), A063092(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively. - Philippe Deléham, Jan 14 2012
T(n,k) = T(n-1,k) + T(n-2,k-1) with T(0,0)=1, T(1,0)=2, T(1,1)=0 and T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Jan 14 2012

A259775 Stepped path in P(k,n) array of k-th partial sums of squares (A000290).

Original entry on oeis.org

1, 5, 6, 20, 27, 77, 112, 294, 450, 1122, 1782, 4290, 7007, 16445, 27456, 63206, 107406, 243542, 419900, 940576, 1641486, 3640210, 6418656, 14115100, 25110020, 54826020, 98285670, 213286590, 384942375

Offset: 1

Luciano Ancora, Jul 05 2015

The term "stepped path" in the name field is the same used in A001405.

Interleaving of terms of the sequences A220101 and A129869. - Michel Marcus, Jul 05 2015

			The array of k-th partial sums of squares begins:
[1], [5],  14,   30,    55,     91,  ...  A000330
1,   [6], [20],  50,   105,    196,  ...  A002415
1,    7,  [27], [77],  182,    378,  ...  A005585
1,    8,   35, [112], [294],   672,  ...  A040977
1,    9,   44,  156,  [450], [1122], ...  A050486
1,   10,   54,  210,   660,  [1782], ...  A053347
This is essentially A110813 without its first two columns.

Cf. A000330, A002415, A005585, A040977, A050486, A053347.

Table[DifferenceRoot[Function[{a, n}, {(-9168 - 14432*n - 8412*n^2 - 2152*n^3 - 204*n^4)*a[n] +(-1332 - 1902*n - 792*n^2 - 102*n^3)*a[1 + n] + (2100 + 3884*n + 2493*n^2 + 640*n^3 + 51*n^4)*a[2 + n] == 0, a[1] == 1 , a[2] == 5}]][n], {n, 29}]

Conjecture: -(n+5)*(13*n-11)*a(n) +(8*n^2+39*n-35)*a(n-1) +2*(26*n^2+48*n+25)*a(n-2) -4*(8*n+5)*(n-1)*a(n-3)=0. - R. J. Mathar, Jul 16 2015

A266561 12-dimensional square numbers.

Original entry on oeis.org

1, 14, 104, 546, 2275, 8008, 24752, 68952, 176358, 419900, 940576, 1998724, 4056234, 7904456, 14858000, 27041560, 47805615, 82317690, 138389160, 227613750, 366913365, 580610160, 903171360, 1382805840, 2086129500, 3104160696, 4559958144, 6618272584

Offset: 0

Antal Pinter, Dec 31 2015

2*a(n) is number of ways to place 11 queens on an (n+11) X (n+11) chessboard so that they diagonally attack each other exactly 55 times. The maximal possible attack number, p=binomial(k,2)=55 for k=11 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form the corresponding complete graph.

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 p. 15.
OEIS Wiki, Square hyperpyramidal numbers, line d=12 of first table.

Cf. A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.

[Binomial(n+11,11)*(n+6)/6: n in [0..40]]; // Vincenzo Librandi, Jan 01 2016

CoefficientList[Series[(1 + x)/(1 - x)^13, {x, 0, 33}], x] (* Vincenzo Librandi, Jan 01 2016 *)

a(n) = binomial(n+11,11)*(n+6)/6.
a(n) = 2*binomial(n+12,12) - binomial(n+11,11).
a(n) = binomial(n+11,11) + 2*binomial(n+11,12) for n>0.
G.f.: (1+x)/(1-x)^13. - Vincenzo Librandi, Jan 01 2016

A271567 Convolution of nonzero triangular numbers (A000217) and nonzero tetradecagonal numbers (A051866).

Original entry on oeis.org

1, 17, 87, 287, 742, 1638, 3234, 5874, 9999, 16159, 25025, 37401, 54236, 76636, 105876, 143412, 190893, 250173, 323323, 412643, 520674, 650210, 804310, 986310, 1199835, 1448811, 1737477, 2070397, 2452472, 2888952, 3385448, 3947944, 4582809, 5296809, 6097119

Offset: 0

Ilya Gutkovskiy, Apr 12 2016

More generally, the ordinary generating function for the convolution of triangular numbers and k-gonal numbers is (1 + (k - 3)*x)/(1 - x)^6.

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Triangular Number
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Cf. A000217, A051866.

Cf. similar sequences of the convolution of triangular numbers and k-gonal numbers: A005585 (k=4), A051836 (k=5), A034263 (k=6), A027800 (k=7), A051843 (k=8), A051877 (k=9), A051878 (k=10), A051879 (k=11), A051880 (k=12), A056118 (k=13), this sequence (k=14).

/* From definition: */ P:=func; /*, where P(n, k) is the n-th k-gonal number, */ [&+[P(n+1-i, 3)*P(i, 14): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 18 2016

[(n+1)*(n+2)*(n+3)*(n+4)*(12*n+5)/120: n in [0..40]]; // Bruno Berselli, Apr 18 2016

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 17, 87, 287, 742, 1638}, 40]
Table[(n + 1) (n + 2) (n + 3) (n + 4) (12 n + 5)/120, {n, 0, 40}]

O.g.f.: (1 + 11*x)/(1 - x)^6.
E.g.f.: (120 + 1920*x + 3240*x^2 + 1520*x^3 + 245*x^4 + 12*x^5)*exp(x)/120.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = (n + 1)*(n + 2)*(n + 3)*(n + 4)*(12*n + 5)/120.
Sum_{n>=0} 1/a(n) = 20*((15552*(6*log(2) + 3*log(3) + 2*sqrt(3)*log(2 - sqrt(3)) + (2 - sqrt(3))*Pi) - 29449)/531867) = 1.07654258697...

Edited by Bruno Berselli, Apr 18 2016

A271663 Convolution of nonzero squares (A000290) with nonzero pentagonal numbers (A000326).

Original entry on oeis.org

1, 9, 41, 131, 336, 742, 1470, 2682, 4587, 7447, 11583, 17381, 25298, 35868, 49708, 67524, 90117, 118389, 153349, 196119, 247940, 310178, 384330, 472030, 575055, 695331, 834939, 996121, 1181286, 1393016, 1634072, 1907400, 2216137, 2563617, 2953377, 3389163, 3874936

Offset: 0

Ilya Gutkovskiy, Apr 12 2016

More generally, the ordinary generating function for the convolution of nonzero h-gonal numbers and k-gonal numbers is (1 + (h - 3)*x)*(1 + (k - 3)*x)/(1 - x)^6.

G. C. Greubel, Table of n, a(n) for n = 0..1000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Pentagonal Number
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Cf. A000290, A000326.
Cf. A005585: convolution of nonzero squares with nonzero triangular numbers.
Cf. A033455: convolution of nonzero squares with themselves.
Cf. A051836 (after 0): convolution of nonzero triangular numbers with nonzero pentagonal numbers.

/* From definition: */ P:=func; /*, where P(n,k) is the n-th k-gonal number, */ [&+[P(n+1-i,4)*P(i,5): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 12 2016

[(n+1)*(n+2)*(n+3)*(6*n^2+19*n+20)/120: n in [0..40]]; // Bruno Berselli, Apr 12 2016

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 9, 41, 131, 336, 742}, 40]
Table[(n + 1) (n + 2) (n + 3) (6 n^2 + 19 n + 20)/120, {n, 0, 40}]
With[{nmax = 50}, CoefficientList[Series[(120 + 960*x + 1440*x^2 + 680*x^3 + 115*x^4 + 6*x^5)*Exp[x]/120, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 07 2017 *)

vector(40, n, n--; (n+1)*(n+2)*(n+3)*(6*n^2+19*n+20)/120) \\ Altug Alkan, Apr 12 2016

O.g.f.: (1 + x)*(1 + 2*x)/(1 - x)^6.
E.g.f.: (120 + 960*x + 1440*x^2 + 680*x^3 + 115*x^4 + 6*x^5)*exp(x)/120.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = (n + 1)*(n + 2)*(n + 3)*(6*n^2 + 19*n + 20)/120.
Sum_{n>=0} 1/a(n) = 1.149165731...

Edited by Bruno Berselli, Apr 12 2016

A339355 Maximum number of copies of a 12345 permutation pattern in an alternating (or zig-zag) permutation of length n + 7.

Original entry on oeis.org

8, 16, 64, 112, 272, 432, 832, 1232, 2072, 2912, 4480, 6048, 8736, 11424, 15744, 20064, 26664, 33264, 42944, 52624, 66352, 80080, 99008, 117936, 143416, 168896, 202496, 236096, 279616, 323136, 378624, 434112, 503880, 573648, 660288, 746928, 853328, 959728, 1089088, 1218448

Offset: 1

Lara Pudwell, Dec 01 2020

nonn

The maximum number of copies of 123 in an alternating permutation is motivated in the Notices reference, and the argument here is analogous.

			a(1) = 8. The alternating permutation of length 1 + 7 = 8 with the maximum number of copies of 12345 is 13254768. The eight copies are 12468, 12478, 12568, 12578, 13468, 13478, 13568, and 13578.

Georg Fischer, Table of n, a(n) for n = 1..200
Lara Pudwell, From permutation patterns to the periodic table, Notices of the American Mathematical Society. 67.7 (2020), 994-1001.

Cf. A005585, A033455, A168380.

a := proc(n2) local n; n:= floor(n2/2): if n2 = 2*n then 32*binomial(n+4,5) - 16*binomial(n+3,4) else n:=n+1; (4*n*(n^4+5*n^3+10*n^2+10*n+4))/15 fi end; seq(a(n), n=1..20); # Georg Fischer, Nov 25 2022

a(2*n) = 16*A005585(n) = 32*binomial(n+4, 5) - 16*binomial(n+3, 4).
a(2*n-1) = 8*A033455(n) = (4*n*(n^4 + 5*n^3 + 10*n^2 + 10*n + 4))/15.
D-finite with recurrence: (n-1)*((n-3)^2+9*n-6)*a(n) - (2*(n-3)^2+20*n-16)*a(n-1) - (n+5)*((n-3)^2+11*n-2)*a(n-2) = 0. - Georg Fischer, Nov 25 2022

cp's OEIS Frontend

A213561 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = m*(m+1)/2, m=n-1+h, n>=1, h>=1, and ** = convolution.

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A067056 a(n) = (1)*(2 + 3 + 4 + ... + n) + (1 + 2)*(3 + 4 + 5 + ... + n) + (1 + 2 + 3)*(4 + 5 + 6 + ... + n) + ... + (1 + 2 + 3 + ... + n-1)*n.

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A207543 Triangle read by rows, expansion of (1+y*x)/(1-2*y*x+y*(y-1)*x^2).

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A104711 Triangle T(n,m) = sum_{k=m..n} A001263(k,m).

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A129710 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.

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A259775 Stepped path in P(k,n) array of k-th partial sums of squares (A000290).

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A266561 12-dimensional square numbers.

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A067056 a(n) = (1)(2 + 3 + 4 + ... + n) + (1 + 2)(3 + 4 + 5 + ... + n) + (1 + 2 + 3)(4 + 5 + 6 + ... + n) + ... + (1 + 2 + 3 + ... + n-1)n.

A207543 Triangle read by rows, expansion of (1+yx)/(1-2y*x+y(y-1)x^2).