A213751 -id:A213751 - cp's OEIS frontend

A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90

Offset: 1

Clark Kimberling, Jun 14 2012

Principal diagonal: A002412.
Antidiagonal sums: A002415.
Row 1:  (1,2,3,...)**(1,2,3,...) = A000292.
Row 2:  (1,2,3,...)**(2,3,4,...) = A005581.
Row 3:  (1,2,3,...)**(3,4,5,...) = A006503.
Row 4:  (1,2,3,...)**(4,5,6,...) = A060488.
Row 5:  (1,2,3,...)**(5,6,7,...) = A096941.
Row 6:  (1,2,3,...)**(6,7,8,...) = A096957.
...
In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples:  let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.
...
In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.
b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
h .......... h .......... A213500 . A002412 . A002415
h .......... h^2 ........ A212891 . A213436 . A024166
h^2 ........ h .......... A213503 . A117066 . A033455
h^2 ........ h^2 ........ A213505 . A213546 . A213547
h .......... h*(h+1)/2 .. A213548 . A213549 . A051836
h*(h+1)/2 .. h .......... A213550 . A002418 . A005585
h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923
h .......... h^3 ........ A213553 . A213554 . A101089
h^3 ........ h .......... A213555 . A213556 . A213547
h^3 ........ h^3 ........ A213558 . A213559 . A213560
h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563
h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094
2^(h-1) .... h .......... A213568 . A213569 . A047520
2^(h-1) .... h^2 ........ A213573 . A213574 . A213575
h .......... Fibo(h) .... A213576 . A213577 . A213578
Fibo(h) .... h .......... A213579 . A213580 . A053808
Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988
Fibo(h+1) .. h .......... A213584 . A213585 . A213586
Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589
h^2 ........ Fibo(h) .... A213590 . A213504 . A213557
Fibo(h) .... h^2 ........ A213566 . A213567 . A213570
h .......... -1+2^h ..... A213571 . A213572 . A213581
-1+2^h ..... h .......... A213582 . A213583 . A156928
-1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749
h .......... 2*h-1 ...... A213750 . A007585 . A002417
2*h-1 ...... h .......... A213751 . A051662 . A006325
2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238
2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755
-1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758
2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764
2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767
Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770
Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776
Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881
h .......... 1+[h/2] .... A213778 . A213779 . A213780
1+[h/2] .... h .......... A213781 . A213782 . A005712
1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760
h .......... 3*h-2 ...... A213761 . A172073 . A002419
3*h-2 ...... h .......... A213771 . A213772 . A132117
3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818
h .......... 3*h-1 ...... A213819 . A213820 . A153978
3*h-1 ...... h .......... A213821 . A033431 . A176060
3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824
3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827
3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830
2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560
3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834
h .......... 4*h-3 ...... A213835 . A172078 . A051797
4*h-3 ...... h .......... A213836 . A213837 . A071238
4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840
2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843
2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846
4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324
[(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850
h .......... C(2*h-2,h-1) A213853
...
Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.

			Northwest corner (the array is read by southwest falling antidiagonals):
  1,  4, 10, 20,  35,  56,  84, ...
  2,  7, 16, 30,  50,  77, 112, ...
  3, 10, 22, 40,  65,  98, 140, ...
  4, 13, 28, 50,  80, 119, 168, ...
  5, 16, 34, 60,  95, 140, 196, ...
  6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.

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

Cf. A000027.

b[n_] := n; c[n_] := n
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}]  (* A213500 *)

t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017

def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
    print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017

T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
T(n,k) = 2*T(n-1,k) - T(n-2,k).
G.f. for row n:  x*(n - (n - 1)*x)/(1 - x)^4.

A006325 4-dimensional analog of centered polygonal numbers.

Original entry on oeis.org

0, 0, 1, 7, 26, 70, 155, 301, 532, 876, 1365, 2035, 2926, 4082, 5551, 7385, 9640, 12376, 15657, 19551, 24130, 29470, 35651, 42757, 50876, 60100, 70525, 82251, 95382, 110026, 126295, 144305, 164176, 186032, 210001, 236215, 264810, 295926

Offset: 0

Albert Rich (Albert_Rich(AT)msn.com)

If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-4) is equal to the number of 6-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
For n>0, a(n+1) is the n-th antidiagonal sum of A213751. - Clark Kimberling, Jun 20 2012
This sequence is the case m=n-1, k=n+3 of b(m,k) = m*(m+1)*((k-2)*m-(k-5))/6, which is the m-th k-gonal pyramidal number. - Luciano Ancora, Apr 11 2015
Starting (1, 7, 26, ...), this is the binomial transform of (1, 6, 13, 12, 4, 0, 0, 0, ...). - Gary W. Adamson, Jul 31 2015
Also starting (1, 7, 26, ...), this appears to be the number of magic labelings of the cycle-of-loops graph LOOP X C_4 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph. - David J. Seal, Sep 13 2017
The conjecture by David J. Seal is true and easily proved using MacMahon's Omega operators via the "Omega" package for Mathematica authored by Axel Riese (obtaining (up to an offset) the generating function listed in the formula section below). See the second Mathematica program in which the edges of LOOP X C_4 are indexed as in the example below. The Omega package can be downloaded from the link provided in the article by G. E. Andrews et al. - L. Edson Jeffery, Oct 15 2017

			A representation of the LOOP X C_4 graph, with edges and loops indexed as shown, as used in the second Mathematica program below:
.             3         1
.              O_______O
.              |   2   |
.              |4     0|
.              |_______|
.              O   6   O
.             5         7

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
David Galvin and Courtney Sharpe, Independent set sequence of linear hyperpaths, arXiv:2409.15555 [math.CO], 2024. See p. 7.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
J. K. Merikoski, R. Kumar and R. A. Rajput, Upper bounds for the largest eigenvalue of a bipartite graph, Electronic Journal of Linear Algebra ISSN 1081-3810, A publication of the International Linear Algebra Society, Volume 26, pp. 168-176, April 2013.
Richard P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005900, A169938, A213751, A288486, A019298, A061907, A244497, A244879, A244873, A244880.
Cf. A000027, A000217, A019298, A244497, A244879, A244873, A244880, A293310, A293309 (magic labelings of LOOP X C_k, for k = 1..3,5..10).
Cf. A293311, A293312.

[n*(n-1)*(n^2-n+1)/6: n in [0..40]]; // Vincenzo Librandi, May 22 2011

Table[n*(n-1)*(n^2-n+1)/6, {n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)
<< Omega.m; n = 4; cond = {}; Do[AppendTo[cond, Sum[a[Mod[2*k - j, 2*n]], {j, 0, 2}] == a[2*n]], {k, 0, n - 1}]; f = OEqSum[Product[x[i]^a[i], {i, 0, 2*n}], cond, u][[1]]; Do[f = OEqR[f, Subscript[u, k]], {k, n}];
(* Generating function: *)
f = Factor[f /. {x[2*n] -> x} /. {x[_] -> 1}]
(* This sequence (with initial zeros dropped): *)
CoefficientList[Series[f, {x, 0, 35}], x] (* L. Edson Jeffery, Oct 15 2017 *)

a(n)=n*(n-1)*(n^2-n+1)/6 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = n*(n-1)*(n^2-n+1)/6.
a(n) = ((n^5 - (n-1)^5) - (n^1 - (n-1)^1))/30 = (n^5 - (n-1)^5 - 1)/30. - Xavier Acloque, Jan 25 2003
The partial sums of the octahedral numbers: a(n+1) = Sum_{i=0..n} A005900(i). - Jonathan Vos Post, Mar 14 2006
G.f.: -x^2*(x+1)^2/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} min(i,j)^2. - Enrique Pérez Herrero, Jan 15 2013 [Which is just rephrasing the partial sum formula with the Murthy formula in A005900. - R. J. Mathar, Jun 14 2014]
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 4. - Yosu Yurramendi, Sep 03 2013
From Daniel Poveda Parrilla, Sep 09 2017: (Start)
a(n) = A169938(n-1)/6.
a(n+1) = A288486(n)/24. (End)
Sum_{n>=2} 1/a(n) = 12 - 2 * sqrt(3) * tanh(sqrt(3)*Pi/2). - Amiram Eldar, Jun 28 2020
E.g.f.: exp(x)*x^2*(3 + 4*x + x^2)/6. - Stefano Spezia, Dec 12 2021

A101986 Maximum sum of products of successive pairs in a permutation of order n+1.

Original entry on oeis.org

0, 2, 9, 23, 46, 80, 127, 189, 268, 366, 485, 627, 794, 988, 1211, 1465, 1752, 2074, 2433, 2831, 3270, 3752, 4279, 4853, 5476, 6150, 6877, 7659, 8498, 9396, 10355, 11377, 12464, 13618, 14841, 16135, 17502, 18944, 20463, 22061, 23740, 25502

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Jan 29 2005

1 3 5 4 2 is the 11th permutation, in lexical order. of order 5. Its reverse 2 4 5 3 1 is the 41st. The earliest permutation of order 6 is the 41st, 1 3 5 6 4 2. This pattern continues as far as I have looked, so its reversal 2 4 6 5 3 1 is the 191st and the earliest permutation of order 7 is the 191st, et cetera.
Comments from Dmitry Kamenetsky, Dec 15 2006: (Start)
This sequence is related to A026035, except here we take the maximum sum of products of successive pairs. Here is a method for generating such permutations. Start with two lists, the first has numbers 1 to n, while the second is empty.
Repeat the following operations until the first list is empty: 1. Move the smallest number of the first list to the leftmost available position in the second list. The move operation removes the original number from the first list. 2. Move the smallest number of the first list to the rightmost available position in the second list. For example when n=8, the permutation is 1, 3, 5, 7, 8, 6, 4, 2. (End)
Convolution of odd numbers and integers greater than 1. - Reinhard Zumkeller, Mar 30 2012
For n>0, a(n) is row 2 of the convolution array A213751. - Clark Kimberling, Jun 20 2012

			The permutations of order 5 with maximum sum of products is 1 3 5 4 2 and its reverse, since (1*3)+(3*5)+(5*4)+(4*2) is 46. All others are empirically less than 46. So a(4) = 46.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Leroy Quet, Max/Min Sums From Permutations - a message on Jan 28 2005 to the sequence list, asking for someone to provide more values and to submit these to OEIS.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Pairwise sums of A005581.

Cf. A000290, A000330, A008865, A026035, A160805.

a101986 n = sum $ zipWith (*) [1,3..] (reverse [2..n+1])
-- Reinhard Zumkeller, Mar 30 2012

0 1 9 2 & p. % 6 & p. (A) NB. the polynomial P such that P(n) is a(n).
NB. where 0 1 9 2 are the coefficients in ascending order of the numerator of a rational polynomial and 6 is the (constant) coefficient of its denominator. J's primitive function p. produces a polynomial with these coefficients. Division is indicated by % . Thus the J expression (A) is equivalent to the formula above.

a:=n->add((n+j^2),j=1..n): seq(a(n),n=0..41); # Zerinvary Lajos, Jul 27 2006

Table[(n + 9 n^2 + 2 n^3)/6, {n, 0, 41}] (* Robert G. Wilson v, Feb 04 2005 *)

a(n)=n*(2*n^2+9*n+1)/6 \\ Charles R Greathouse IV, Jan 17 2012

a(n) = n*(2*n^2 + 9*n + 1)/6.
a(n+1) = a(n) + A008865(n+2); a(n) = A160805(n) - 4. [Reinhard Zumkeller, May 26 2009]
G.f.: x*(1+x)*(2-x)/(1-x)^4. - L. Edson Jeffery, Jan 17 2012
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3, a(0)=0, a(1)=2, a(2)=9, a(3)=23. - L. Edson Jeffery, Jan 17 2012
a(n) = A000330(n) + A005449(n) - A000217(n). - Richard R. Forberg, Aug 07 2013
a(n) = 1 + sum( A008865(i), i=1..n+1 ). [Bruno Berselli, Jan 13 2015]
a(n) = A000290(n) + A000330(n). - J. M. Bergot, Apr 26 2018

Edited by Bruno Berselli, Jan 13 2015

Name edited by Alois P. Heinz, Feb 02 2019

A051662 House numbers: a(n) = (n+1)^3 + Sum_{i=1..n} i^2.

Original entry on oeis.org

1, 9, 32, 78, 155, 271, 434, 652, 933, 1285, 1716, 2234, 2847, 3563, 4390, 5336, 6409, 7617, 8968, 10470, 12131, 13959, 15962, 18148, 20525, 23101, 25884, 28882, 32103, 35555, 39246, 43184, 47377, 51833, 56560, 61566, 66859, 72447, 78338, 84540, 91061, 97909

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Binomial transform of [1, 8, 15, 8, 0, 0, 0, ...]. - Gary W. Adamson, Nov 23 2007

Principal diagonal of the convolution array A213751. - Clark Kimberling, Jun 20 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Josh Deprez, Fair amenability for semigroups, arXiv preprint arXiv:1310.5589 [math.GR], 2013-2015.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A220084 (for a list of numbers of the form n*P(k,n) - (n-1)*P(k,n-1), where P(k,n) is the n-th k-gonal pyramidal number).

Cf. A000330, A000578, A007318, A023855, A213751.

- following Gary W. Adamson's comment.
a051662 = sum . zipWith (*) [1, 8, 15, 8] . a007318_row
-- Reinhard Zumkeller, Feb 19 2015

a:=n->sum(k^2, k=1..n):seq(a(n)+sum(n^2, k=2..n), n=1...40); # Zerinvary Lajos, Jun 11 2008

Table[(n+1)^3+Sum[i^2,{i,n}],{n,0,40}] (* or *) LinearRecurrence[ {4,-6,4,-1}, {1,9,32,78},40] (* Harvey P. Dale, Jun 23 2011 *)

A051662(n):=((8*n+21)*n+19)*n/6+1$ makelist(A051662(n),n,0,15); /* Martin Ettl, Dec 13 2012 */

a(n)=((8*n+21)*n+19)*n/6+1 \\ Charles R Greathouse IV, Jun 23 2011

a(n) = (n+1)*(8*n^2 + 13*n + 6)/6.
a(n) = A000578(n+1) + A000330(n).
From Harvey P. Dale, Jun 23 2011: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(0)=1, a(1)=9, a(2)=32, a(3)=78.
G.f.: (1+5*x+2*x^2)/(x-1)^4. (End)
a(n) = (n+1)*A000330(n+1) - n*A000330(n). - Bruno Berselli, Dec 11 2012
a(n) = A023855(2*n) + A023855(2*n+1). - Luc Rousseau, Feb 24 2018
E.g.f.: exp(x)*(6 + 48*x + 45*x^2 + 8*x^3)/6. - Elmo R. Oliveira, Aug 06 2025

Corrected by T. D. Noe, Nov 01 2006 and Nov 08 2006

cp's OEIS Frontend

A213500 Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.

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A006325 4-dimensional analog of centered polygonal numbers.

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Magma

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A101986 Maximum sum of products of successive pairs in a permutation of order n+1.

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A051662 House numbers: a(n) = (n+1)^3 + Sum_{i=1..n} i^2.

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Haskell

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A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.