A213761 -id:A213761 - 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.

A002419 4-dimensional figurate numbers: a(n) = (6n-2)binomial(n+2,3)/4.

Original entry on oeis.org

1, 10, 40, 110, 245, 476, 840, 1380, 2145, 3190, 4576, 6370, 8645, 11480, 14960, 19176, 24225, 30210, 37240, 45430, 54901, 65780, 78200, 92300, 108225, 126126, 146160, 168490, 193285, 220720, 250976, 284240, 320705, 360570, 404040, 451326, 502645, 558220

Offset: 1

N. J. A. Sloane

a(n) is the n-th antidiagonal sum of the convolution array A213761. - Clark Kimberling, Jul 04 2012
Convolution of A000027 with A000567 (excluding 0). - Bruno Berselli, Dec 07 2012
a(n) = the sum of all the ways of adding the k-tuples of A016777(0) to A016777(n-1). For n=4, the terms are 1,4,7,10 giving (1)+(4)+(7)+(10)=22; (1+4)+(4+7)+(7+10)=33; (1+4+7)+(4+7+10)=33; (1+4+7+10)=22; adding 22+33+33+22=110. - J. M. Bergot, Jun 26 2017
Also the number of chordless cycles in the (n+2)-crown graph. - Eric W. Weisstein, Jan 02 2018

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Eric Weisstein's World of Mathematics, Chordless Cycle.
Eric Weisstein's World of Mathematics, Crown Graph.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index to sequences related to pyramidal numbers.

Cf. A093563 ((6, 1) Pascal, column m=4).
Cf. A000027, A000567, A002414 (first differences), A016777, A080852, A213761.
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

List([1..40], n-> n*(n+1)*(n+2)*(3*n-1)/12); # G. C. Greubel, Jul 03 2019

/* A000027 convolved with A000567 (excluding 0): */ A000567:=func; [&+[(n-i+1)*A000567(i): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Dec 07 2012

CoefficientList[Series[(1+5*x)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Jun 20 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 10, 40, 110, 245}, 40] (* Harvey P. Dale, Nov 30 2014 *)
Table[n(n+1)(n+2)(3n-1)/12, {n, 40}] (* Eric W. Weisstein, Jan 02 2018 *)
Table[Sum[2 x + 3 x^2 - 2 y, {x, 0, g}, {y, x, g}], {g, 1, 20}] (* Horst H. Manninger, Jun 20 2025 *)

a(n)=(3*n-1)*binomial(n+2,3)/2 \\ Charles R Greathouse IV, Sep 24 2015

A002419_list, m = [], [6, 1, 1, 1, 1]
for _ in range(10**2):
    A002419_list.append(m[-1])
    for i in range(4):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

[n*(n+1)*(n+2)*(3*n-1)/12 for n in (1..40)] # G. C. Greubel, Jul 03 2019

a(n) = (3*n-1)*binomial(n+2, 3)/2.
G.f.: x*(1+5*x)/(1-x)^5. - Simon Plouffe in his 1992 dissertation.
Sum_{n>=1} 1/a(n) = (-24+81*log(3) -9*Pi*sqrt(3))/14 = 1.143929... - R. J. Mathar, Mar 29 2011
a(n) = (3*n^4 + 8*n^3 + 3*n^2 - 2*n)/12. - Chai Wah Wu, Jan 24 2016
a(n) = A080852(6,n-1). - R. J. Mathar, Jul 28 2016
E.g.f.: x*(12 + 48*x + 26*x^2 + 3*x^3)*exp(x)/12. - G. C. Greubel, Jul 03 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(3*sqrt(3)*Pi - 32*log(2) + 8)/7. - Amiram Eldar, Feb 11 2022

A077414 a(n) = n(n - 1)(n + 2)/2.

Original entry on oeis.org

0, 4, 15, 36, 70, 120, 189, 280, 396, 540, 715, 924, 1170, 1456, 1785, 2160, 2584, 3060, 3591, 4180, 4830, 5544, 6325, 7176, 8100, 9100, 10179, 11340, 12586, 13920, 15345, 16864, 18480, 20196, 22015, 23940, 25974, 28120, 30381, 32760, 35260

Offset: 1

Wolfdieter Lang, Nov 29 2002

Number of independent components of a certain 3-tensor in n-space.
a(n) is the number of independent components of a 3-tensor t(a,b,c) which satisfies t(a,b,c) = t(b,a,c) and Sum_{a=1..n} t(a,a,c) = 0 for all c, with a,b,c range 1..n. (3-tensor in n-dimensional space which is symmetric and traceless in one pair of its indices.)
Row 2 of the convolution array A213761.  - Clark Kimberling, Jul 04 2012
Also, the number of ways to place two dominoes horizontally in the same row on an (n+2) X (n+2) chessboard. - Ralf Stephan, Jun 09 2014
Also, the sum of all the numbers in a completely filled n X n tic-tac-toe board with n-1 players using the numbers 0, 1, 2,... n-2. See "Sums of Square Tic Tac Toe Boards that end in a Draw" in links for proof. - Tanner Robeson, Aug 23 2020
a(n) is the number of degrees of freedom in a tetrahedral cell for a Raviart-Thomas finite element space of order n. - Matthew Scroggs, Jan 02 2021

			For n=6, a(6) = 1*(3*5+1)+2*(3*4+1)+3*(3*3+1)+4*(3*2+1)+5*(3*1+1) = 120. - _Bruno Berselli_, Feb 13 2014
G.f. = 4*x^2 + 15*x^3 + 36*x^4 + 70*x^5 + 120*x^6 + 189*x^7 + 280*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
DefElement, Raviart-Thomas
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Tanner Robeson, Sums of Square Tic Tac Toe Boards that end in a Draw.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Index to sequences related to polygonal numbers.

Cf. A000096, A005564, A057145, A115067 (first differences), A213761.

Cf. similar sequences of the type m*(m+1)*(m+k)/2 listed in A267370.

[n*(n-1)*(n+2)/2: n in [1..30]]; // G. C. Greubel, Jan 18 2018

A077414:=n->n*(n-1)*(n+2)/2: seq(A077414(n), n=1..60); # Wesley Ivan Hurt, Apr 09 2017

Table[(n (n - 1) (n + 2))/2, {n, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 4, 15, 36}, 50] (* Harvey P. Dale, Jun 04 2012 *)
CoefficientList[Series[x (4 - x)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 14 2014 *)

a(n)=n*(n-1)*(n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015

concat(0, Vec(x^2*(4-x)/(1-x)^4 + O(x^200))) \\ Altug Alkan, Jan 15 2016

a(n) = n * ( binomial(n+1, 2)-1 ).
G.f.: x^2*(4-x)/(1-x)^4.
a(n) = n*Sum_{j=2..n} j. - Zerinvary Lajos, Sep 12 2006
a(1)=0, a(2)=4, a(3)=15, a(4)=36; for n>4, a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Jun 04 2012
a(n) = Sum_{i=1..n-1} i*(3*(n-i)+1). - Bruno Berselli, Feb 13 2014
a(-n) = -A005564(n). - Michael Somos, Jun 09 2014
a(n) = A057145(n,n+2). - R. J. Mathar, Jul 28 2016
a(n) = t(n,t(n,1)) + n, where t(n,k) = n*(n+1)/2 + k*n and t(n,1) = A000096(n). - Bruno Berselli, Feb 28 2017
a(n) = n^3/2 + n^2/2 - n. - Tanner Robeson, Aug 23 2020
Sum_{n>=2} 1/a(n) = 7/18. - Amiram Eldar, Oct 07 2020
Sum_{n>=2} (-1)^n/a(n) = 4*log(2)/3 - 13/18. - Amiram Eldar, Feb 22 2022
E.g.f.: exp(x)*x^2*(4 + x)/2. - Stefano Spezia, Jan 03 2023

A172073 a(n) = (4n^3 + n^2 - 3n)/2.

Original entry on oeis.org

0, 1, 15, 54, 130, 255, 441, 700, 1044, 1485, 2035, 2706, 3510, 4459, 5565, 6840, 8296, 9945, 11799, 13870, 16170, 18711, 21505, 24564, 27900, 31525, 35451, 39690, 44254, 49155, 54405, 60016, 66000, 72369, 79135, 86310, 93906, 101935, 110409, 119340, 128740

Offset: 0

Vincenzo Librandi, Jan 25 2010

14-gonal (or tetradecagonal) pyramidal numbers generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=6.
In fact, the sequence is related to A000567 by a(n) = n*A000567(n) - Sum_{i=0..n-1} A000567(i) and this is the case d=6 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Nov 29 2010
Except for the initial 0, this is the principal diagonal of the convolution array A213761. - Clark Kimberling, Jul 04 2012
Starting (1, 15, 54, ...), this is the binomial transform of (1, 14, 25, 12, 0, 0, 0, ...). - Gary W. Adamson, Jul 29 2015

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. - Bruno Berselli, Feb 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
Index to sequences related to pyramidal numbers.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences listed in A237616.

Cf. A000567, A051866, A194454.

List([0..40], n-> n*(n+1)*(4*n-3)/2); # G. C. Greubel, Aug 30 2019

[(4*n^3+n^2-3*n)/2: n in [0..50]]; // Vincenzo Librandi, Jan 01 2014

seq(n*(n+1)*(4*n-3)/2, n=0..40); # G. C. Greubel, Aug 30 2019

f[n_]:= n(n+1)(4n-3)/2; Array[f, 40, 0]
LinearRecurrence[{4,-6,4,-1},{0,1,15,54},40] (* Harvey P. Dale, Jan 29 2013 *)
CoefficientList[Series[x (1+11x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 01 2014 *)

a(n)=(4*n^3+n^2-3*n)/2 \\ Charles R Greathouse IV, Oct 07 2015

[n*(n+1)*(4*n-3)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019

a(n) = n*(n+1)*(4*n-3)/2.
From Bruno Berselli, Dec 15 2010: (Start)
G.f.: x*(1+11*x)/(1-x)^4.
a(n) = Sum_{i=0..n} A051866(i). (End)
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)=1, a(2)=15, a(3)=54. - Harvey P. Dale, Jan 29 2013
a(n) = Sum_{i=0..n-1} (n-i)*(12*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*Pi/21 + 8*log(2)/7 - 2/7.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(2)*Pi/21 + 8*sqrt(2)*log(sqrt(2)+2)/21 - (20 + 4*sqrt(2))*log(2)/21 + 2/7. (End)
E.g.f.: exp(x)*x*(2 + 13*x + 4*x^2)/2. - Elmo R. Oliveira, Aug 04 2025

Edited by Bruno Berselli, Dec 14 2010

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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A002419 4-dimensional figurate numbers: a(n) = (6*n-2)*binomial(n+2,3)/4.

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A077414 a(n) = n*(n - 1)*(n + 2)/2.

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A172073 a(n) = (4*n^3 + n^2 - 3*n)/2.

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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.

A002419 4-dimensional figurate numbers: a(n) = (6n-2)binomial(n+2,3)/4.

A077414 a(n) = n(n - 1)(n + 2)/2.

A172073 a(n) = (4n^3 + n^2 - 3n)/2.