A130748 -id:A130748 - 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.

A260260 a(n) = n(16n^2 - 21*n + 7)/2.

Original entry on oeis.org

0, 1, 29, 132, 358, 755, 1371, 2254, 3452, 5013, 6985, 9416, 12354, 15847, 19943, 24690, 30136, 36329, 43317, 51148, 59870, 69531, 80179, 91862, 104628, 118525, 133601, 149904, 167482, 186383, 206655, 228346, 251504, 276177, 302413, 330260, 359766, 390979

Offset: 0

Bruno Berselli, Jul 21 2015

Similar sequences, where P(s, m) = ((s-2)*m^2-(s-4)*m)/2 is the m-th s-gonal number:
A000578: P(3, m)*P( 3, m) - P(3, m-1)*P( 3, m-1);
A213772: P(3, m)*P( 4, m) - P(3, m-1)*P( 4, m-1) for m>0;
A005915: P(3, m)*P( 5, m) - P(3, m-1)*P( 5, m-1)   "    ;
A130748: P(3, m)*P( 6, m) - P(3, m-1)*P( 6, m-1) for m>1;
A027849: P(3, m)*P( 7, m) - P(3, m-1)*P( 7, m-1) for m>0;
A214092: P(3, m)*P( 8, m) - P(3, m-1)*P( 8, m-1)   "    ;
A100162: P(3, m)*P( 9, m) - P(3, m-1)*P( 9, m-1)   "    ;
A260260: P(3, m)*P(10, m) - P(3, m-1)*P(10, m-1), this sequence;
A100165: P(3, m)*P(11, m) - P(3, m-1)*P(11, m-1) for m>0.

Bruno Berselli, Table of n, a(n) for n = 0..1000
OEIS Wiki, Figurate numbers.
Wikipedia, Polygonal numbers: Table of values.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A000292, A001107.
Subsequence of A047275.
Sequences of the same type (see comment): A000578, A005915, A027849, A100162, A100165, A130748, A213772, A214092.

Magma
```
[n*(16*n^2-21*n+7)/2: n in [0..40]];
```

Table[n (16 n^2 - 21 n + 7)/2, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{0,1,29,132},40] (* Harvey P. Dale, May 08 2025 *)

vector(40, n, n--; n*(16*n^2-21*n+7)/2)

Sage
```
[n*(16*n^2-21*n+7)/2 for n in (0..40)]
```

G.f.: x*(1 + 25*x + 22*x^2)/(1 - x)^4. [corrected by Georg Fischer, May 10 2019]
a(n) = A000217(n)*A001107(n) - A000217(n-1)*A001107(n-1), with A000217(-1) = 0.
a(n) = A000292(n) + 25*A000292(n-1) + 22*A000292(n-2), with A000292(-2) = A000292(-1) = 0.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Wesley Ivan Hurt, Dec 18 2020
E.g.f.: exp(x)*x*(2 + 27*x + 16*x^2)/2. - Elmo R. Oliveira, Aug 08 2025

A220084 a(n) = (n + 1)(20n^2 + 19*n + 6)/6.

Original entry on oeis.org

1, 15, 62, 162, 335, 601, 980, 1492, 2157, 2995, 4026, 5270, 6747, 8477, 10480, 12776, 15385, 18327, 21622, 25290, 29351, 33825, 38732, 44092, 49925, 56251, 63090, 70462, 78387, 86885, 95976, 105680, 116017, 127007, 138670, 151026, 164095, 177897, 192452

Offset: 0

Bruno Berselli, Dec 11 2012

Sequence related to heptagonal pyramidal numbers (A002413) by a(n) = n*A002413(n) - (n-1)*A002413(n-1).
Other sequences of numbers of the form m*P(k,m)-(m-1)*P(k,m-1), where P(k,m) is the m-th k-gonal pyramidal number:
k=3, A002412(m) = m*A000292(m)-(m-1)*A000292(m-1);
k=4, A051662(m) = (m+1)*A000330(m+1)-m*A000330(m);
k=5, A213772(m) = m*A002411(m)-(m-1)*A002411(m-1);
k=6, A213837(m) = m*A002412(m)-(m-1)*A002412(m-1);
k=7, this sequence;
k=8, A130748(m) = m*A002414(m)-(m-1)*A002414(m-1).
Also, first bisection of A212983.
Binomial transform of (1, 14, 33, 20, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A000330, A000566, A002411, A002412, A002413, A002414, A051662, A130748, A212983, A213772, A213837.

[(n+1)*(20*n^2+19*n+6)/6: n in [0..40]]; // Bruno Berselli, Jun 28 2016

/* By first comment: */  A002413:=func; [n*A002413(n)-(n-1)*A002413(n-1): n in [1..40]];

I:=[1,15,62,162]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Table[(n + 1) (20 n^2 + 19 n + 6)/6, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{1,15,62,162},40] (* Harvey P. Dale, Dec 23 2012 *)
CoefficientList[Series[(1 + 11 x + 8 x^2) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

makelist((n+1)*(20*n^2+19*n+6)/6, n, 0, 20); /* Martin Ettl, Dec 12 2012 */

a(n)=(n+1)*(20*n^2+19*n+6)/6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1+11*x+8*x^2)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), for n>3, a(0)=1, a(1)=15, a(2)=62, a(3)=162. - Harvey P. Dale, Dec 23 2012
a(n) = (n+1)*A000566(n+1) + Sum_{i=0..n} A000566(i). - Bruno Berselli, Dec 18 2013
E.g.f.: exp(x)*(6 + 84*x + 99*x^2 + 20*x^3)/6. - Elmo R. Oliveira, Aug 06 2025

A213833 Rectangular array: (row n) = b**c, where b(h) = 3h-2, c(h) = 2n-3+2*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 7, 3, 24, 17, 5, 58, 48, 27, 7, 115, 102, 72, 37, 9, 201, 185, 146, 96, 47, 11, 322, 303, 255, 190, 120, 57, 13, 484, 462, 405, 325, 234, 144, 67, 15, 693, 668, 602, 507, 395, 278, 168, 77, 17, 955, 927, 852, 742, 609, 465

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A103748.
Antidiagonal sums: A213834.
Row 1, (1,3,5,7,...)**(1,3,5,7,...): A081436.
Row 2, (1,3,5,7,...)**(3,5,7,9,...): A144640.
Row 3, (1,3,5,7,...)**(5,7,9,11,...): (2*k^3 + 11*k^2 - 3*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1....7....24....58....115
3....17...48....102...185
5....27...72....146...255
7....37...96....190...325
9....47...120...234...395
11...57...144...278...465

Clark Kimberling, Antidiagonals n = 1..12, flattened

Cf. A212500.

b[n_]:=3n-2;c[n_]:=2n-1;
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}] (* A213833 *)
Table[t[n,n],{n,1,40}] (* A130748 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213834 *)

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

G.f. for row n: f(x)/g(x), where f(x) = x*((2*n-1) + (2*n+1)*x - (4*n-6)*x^2) and g(x) = (1-x)^4.

A302302 Number of triples (i,j,k) such that i+j+k > 0 with -n <= i,j,k <= n.

Original entry on oeis.org

0, 10, 53, 153, 334, 620, 1035, 1603, 2348, 3294, 4465, 5885, 7578, 9568, 11879, 14535, 17560, 20978, 24813, 29089, 33830, 39060, 44803, 51083, 57924, 65350, 73385, 82053, 91378, 101384, 112095, 123535, 135728, 148698, 162469, 177065, 192510, 208828, 226043

Offset: 0

Andres Cicuttin, Apr 05 2018

If we add the triples (i,j,k) such that i+j+k >= 0 then the corresponding numbers seem to be those of A130748.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)

Cf. A003215, A130748.

a[n_]:=Total[Flatten[Table[Table[Table[If[i+j+k>0,1,0],{i,-n,n}],{j,-n,n}],{k,-n,n}]]];
Table[a[n],{n,0,32}]

a(n) = n*(3+9*n+8*n^2)/2; \\ Altug Alkan, Apr 08 2018

From Alois P. Heinz, Apr 08 2018: (Start)
a(n) = n*(3+9*n+8*n^2)/2.
G.f.: (x^2+13*x+10)*x/(x-1)^4.
a(n) = A130748(n) - A003215(n) for n > 0. (End)

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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A260260 a(n) = n*(16*n^2 - 21*n + 7)/2.

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A220084 a(n) = (n + 1)*(20*n^2 + 19*n + 6)/6.

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A213833 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = 2*n-3+2*h, n>=1, h>=1, and ** = convolution.

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A302302 Number of triples (i,j,k) such that i+j+k > 0 with -n <= i,j,k <= n.

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Comments

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Mathematica

PARI

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.

A260260 a(n) = n(16n^2 - 21*n + 7)/2.

A220084 a(n) = (n + 1)(20n^2 + 19*n + 6)/6.

A213833 Rectangular array: (row n) = b**c, where b(h) = 3h-2, c(h) = 2n-3+2*h, n>=1, h>=1, and ** = convolution.