A213752 -id:A213752 - 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.

A100157 Structured rhombic dodecahedral numbers (vertex structure 9).

Original entry on oeis.org

1, 14, 55, 140, 285, 506, 819, 1240, 1785, 2470, 3311, 4324, 5525, 6930, 8555, 10416, 12529, 14910, 17575, 20540, 23821, 27434, 31395, 35720, 40425, 45526, 51039, 56980, 63365, 70210, 77531, 85344, 93665, 102510, 111895, 121836, 132349, 143450, 155155, 167480

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Also structured triakis octahedral numbers (vertex structure 9) (Cf. A100171 = alternate vertex); and structured heptagonal anti-prism numbers (Cf. A100185 = structured anti-prisms).
If Y is a 2-subset of a 2n-set X then, for n>=2, a(n-1) is the number of 4-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007
Let M(2n-1) be a (2n-1)x(2n-1) matrix whose (i,j)-entry equals i^2/(i^2+sqrt(-1)) if i=j and equals 1 otherwise. Then a(n) equals (-1)^(n+1) times the real part of prod(k^2+sqrt(-1),k=1...2n-1) times the determinant of M(2n-1). - John M. Campbell, Sep 07 2011
Principal diagonal of the convolution array A213752. - Clark Kimberling, Jun 20 2012
The Fuss-Catalan numbers are Cat(d,k)= [1/(k*(d-1)+1)]*binomial(k*d,k) and enumerate the number of (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Whieldon and Schuetz link). a(n)= Cat(n,4), so enumerates the number of (n+1)-gon partitions of a (4*(n-1)+2)-gon. Analogous series are A000326 (k=3) and A234043 (k=5). Also, a(n)= A006918(4n+1) = A008610(4n+1) = A053307(4n+1) with offset=0. - Tom Copeland, Oct 05 2014

			For n=4, sum( (4+i)^2, i=-3..3 ) = (4-3)^2+(4-2)^2+(4-1)^2+(4-0)^2+(4+1)^2+(4+2)^2+(4+3)^2 = 140 = a(4). - _Bruno Berselli_, Jul 24 2014

Jolley, Summation of Series, Dover (1961).

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Milan Janjic, Two Enumerative Functions.
A. Schuetz and G. Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
StackExchange, What is a Structured Polyhedron?
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005915 = alternate vertex; A100145 for more on structured polyhedral numbers.

Cf. A000330, A000326, A006918, A008610, A053307, A100171, A100185, A213752, A234043.

[(1/6)*(16*n^3-12*n^2+2*n): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011

with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m*4), m=1..32) ; # Zerinvary Lajos, Jan 02 2008

a(n)=(16*n^3-12*n^2+2*n)/6 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (16*n^3 - 12*n^2 + 2*n)/6.
a(n) = n*(2*n-1)*(4*n-1)/3 = A000330(2*n-1). - Reinhard Zumkeller, Jul 06 2009
Sum_{n>=1} 1/(24*a(n)) = Pi/8-log(2)/2 = 0.046125491418751... [Jolley eq. 251]
G.f.: x*(1+10*x+5*x^2)/(x-1)^4. - R. J. Mathar, Oct 03 2011
a(n) = binomial(2*n+1,3) + binomial(2*n,3). - John Molokach, Jul 10 2013
a(n) = Sum_{i=-(n-1)..(n-1)} (n+i)^2. - Bruno Berselli, Jul 24 2014
From Elmo R. Oliveira, Aug 04 2025: (Start)
E.g.f.: exp(x)*x*(8*x^2 + 18*x + 3)/3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. (End)

A071238 a(n) = n(n+1)(2*n^2+1)/6.

Original entry on oeis.org

0, 1, 9, 38, 110, 255, 511, 924, 1548, 2445, 3685, 5346, 7514, 10283, 13755, 18040, 23256, 29529, 36993, 45790, 56070, 67991, 81719, 97428, 115300, 135525, 158301, 183834, 212338, 244035, 279155, 317936, 360624, 407473, 458745, 514710, 575646, 641839

Offset: 0

N. J. A. Sloane, Jun 12 2002

Binomial transform of [1, 8, 21, 22, 8, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
For n > 0, a(n) is the n-th antidiagonal sum of the convolution arrays A213752 and A213836). - Clark Kimberling, Jun 20 2012
The first differences are given in A277229, as a convolution of the odd-indexed triangular numbers A000217(2*n+1) and the squares A000290(n), n >= 0. - J. M. Bergot, Sep 14 2016

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..2000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000292, A002417, A071270, A277229 (first differences).

[n*(n+1)*(2*n^2+1)/6: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

A071238:=n->n*(n+1)*(2*n^2+1)/6: seq(A071238(n), n=0..60); # Wesley Ivan Hurt, Sep 24 2016

Table[n (n + 1) (2 n^2 + 1)/6, {n, 0, 37}] (* or *)
CoefficientList[Series[x (1 + x) (1 + 3 x)/(1 - x)^5, {x, 0, 37}], x] (* Michael De Vlieger, Sep 14 2016 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,9,38,110},40] (* Harvey P. Dale, Oct 02 2021 *)

a(n)=n*(n+1)*(2*n^2+1)/6; \\ Joerg Arndt, Sep 04 2013

G.f.: x*(1+x)*(1+3*x)/(1-x)^5. - Colin Barker, Mar 22 2012
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4, a(0)=0, a(1)=1, a(2)=9, a(3)=38, a(4)=110. - Yosu Yurramendi, Sep 03 2013
E.g.f.: (1/6)*x*(6 + 21*x + 14*x^2 + 2*x^3)*exp(x). - G. C. Greubel, Sep 17 2016
a(n) = n*A000292(n) + (n-1)*A000292(n-1). - Bruno Berselli, Sep 22 2016
a(n) = A002417(n-1) + A002417(n). - Yasser Arath Chavez Reyes, Feb 15 2024

A143941 The Wiener index of a chain of n triangles (i.e., joined like VVV..VV; here V is a triangle!).

Original entry on oeis.org

3, 14, 37, 76, 135, 218, 329, 472, 651, 870, 1133, 1444, 1807, 2226, 2705, 3248, 3859, 4542, 5301, 6140, 7063, 8074, 9177, 10376, 11675, 13078, 14589, 16212, 17951, 19810, 21793, 23904, 26147, 28526, 31045, 33708, 36519, 39482, 42601, 45880, 49323, 52934

Offset: 1

Emeric Deutsch, Sep 06 2008

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
Row 2 of the convolution array A213752. - Clark Kimberling, Jun 20 2012
Also the circuit rank of the (n+2) X (n+2) bishop graph. - Eric W. Weisstein, May 10 2019

			a(2)=14 because in the graph VV (V is a triangle!) we have 6 distances equal to 1 and 4 distances equal to 2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Bishop Graph
Eric Weisstein's World of Mathematics, Circuit Rank
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A142463, A143940, A213752.

[n*(1+6*n+2*n^2)/3 : n in [1..40]]; // Wesley Ivan Hurt, Apr 08 2015

Maple
```
seq((1/3)*n*(1+6*n+2*n^2), n=1..43);
```

CoefficientList[Series[(3+2*x-x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 03 2012 *)
LinearRecurrence[{4,-6,4,-1},{3,14,37,76},50] (* Harvey P. Dale, Sep 06 2023 *)

a(n) = n*(1 + 6*n + 2*n^2)/3.
G.f.: z*(3 + 2*z - z^2)/(1-z)^4.
a(n) = Sum_{k=1..n} k*A143940(n,k).
a(n) = Sum_{k=1..n} A142463(k). - Richard R. Forberg, Jan 09 2015
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Wesley Ivan Hurt, Apr 08 2015
E.g.f.: exp(x)*x*(9 + 12*x + 2*x^2)/3. - Stefano Spezia, Jan 03 2022

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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A100157 Structured rhombic dodecahedral numbers (vertex structure 9).

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A071238 a(n) = n*(n+1)*(2*n^2+1)/6.

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Magma

Maple

Mathematica

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A143941 The Wiener index of a chain of n triangles (i.e., joined like VVV..VV; here V is a triangle!).

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Crossrefs

Programs

Magma

Maple

Mathematica

Formula

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

A071238 a(n) = n(n+1)(2*n^2+1)/6.