A117066 -id:A117066 - 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.

A185874 Second accumulation array of A051340, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 6, 11, 10, 10, 21, 26, 20, 15, 34, 48, 50, 35, 21, 50, 76, 90, 85, 56, 28, 69, 110, 140, 150, 133, 84, 36, 91, 150, 200, 230, 231, 196, 120, 45, 116, 196, 270, 325, 350, 336, 276, 165, 55, 144, 248, 350, 435, 490, 504, 468, 375, 220, 66, 175, 306, 440, 560, 651, 700, 696, 630, 495, 286, 78, 209, 370, 540, 700, 833, 924, 960, 930, 825, 638, 364, 91, 246, 440, 650, 855, 1036, 1176, 1260, 1275, 1210, 1056, 806, 455, 105, 286, 516, 770, 1025, 1260, 1456, 1596, 1665, 1650, 1540, 1326, 1001, 560

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain: A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
.   1,   3,   6,   10,   15,   21,   28,   36,   45,   55, ...
.   4,  11,  21,   34,   50,   69,   91,  116,  144,  175, ...
.  10,  26,  48,   76,  110,  150,  196,  248,  306,  370, ...
.  20,  50,  90,  140,  200,  270,  350,  440,  540,  650, ...
.  35,  85, 150,  230,  325,  435,  560,  700,  855, 1025, ...
.  56, 133, 231,  350,  490,  651,  833, 1036, 1260, 1505, ...
.  84, 196, 336,  504,  700,  924, 1176, 1456, 1764, 2100, ...
. 120, 276, 468,  696,  960, 1260, 1596, 1968, 2376, 2820, ...
. 165, 375, 630,  930, 1275, 1665, 2100, 2580, 3105, 3675, ...
. 220, 495, 825, 1210, 1650, 2145, 2695, 3300, 3960, 4675, ...
...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

Cf. A144112, A051340, A141419, A185875, A185876.
Row 1 to 5: A000217, A115056, 2*A140096, 10*A000096, 5*A059845.
Column 1 to 3: A000292, A051925, A267370 and 3*A005581.
Main diagonal: A117066.

f[n_, k_] := (1/12)*k*n*(1 + n)*(1 + 3*k + 2*n);
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
Table[f[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten

T(n,k) = k*n*(n+1)*(2*n+3*k+1)/12 for k>=1, n>=1.

Edited by Bruno Berselli, Jan 14 2016

A321500 Triangular table T(n,k) = (n+k)*(n^2+k^2), n >= k >= 0; read by rows n = 0, 1, 2, ...

Original entry on oeis.org

0, 1, 4, 8, 15, 32, 27, 40, 65, 108, 64, 85, 120, 175, 256, 125, 156, 203, 272, 369, 500, 216, 259, 320, 405, 520, 671, 864, 343, 400, 477, 580, 715, 888, 1105, 1372, 512, 585, 680, 803, 960, 1157, 1400, 1695, 2048

Offset: 0

M. F. Hasler, Nov 22 2018

			The table starts:
n | T(n,k), k = 0..n:
0 |   0;
1 |   1,   4;
2 |   8,  15,  32;
3 |  27,  40,  65, 108;
4 |  64,  85, 120, 175, 256;
5 | 125, 156, 203, 272, 369,  500;
6 | 216, 259, 320, 405, 520,  671,  864;
7 | 343, 400, 477, 580, 715,  888, 1105, 1372;
8 | 512, 585, 680, 803, 960, 1157, 1400, 1695, 2048;
etc.

M. F. Hasler, Rows n = 0..141 of triangle, flattened

Cf. A000578 (column 0: the cubes), A033430 (diagonal: 4*n^3), A053698 (column 1).

Cf. A198063 (read as A(n,k)=(n+k)*(n^2+k^2)).

[[(n+k)*(n^2+k^2): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Nov 23 2018

t[n_, k_] := (n + k) (n^2 + k^2); Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 22 2018 *)

A321500(n, k)=(n+k)*(n^2+k^2)
A321500_row(n)=vector(n+1, k, (n+k--)*(n^2+k^2))
A321500_list(N=11)=concat(apply(A321500_row, [0..N]))

[[(n+k)*(n^2+k^2) for k in range(n+1)] for n in range(12)] # G. C. Greubel, Nov 23 2018

Sum_{k=0..n} T(n,k) = 5*n^2*(n+1)*(5*n+1)/12 = 5*A117066(n). - G. C. Greubel, Nov 23 2018

A213503 Rectangular array: (row n) = bc, where b(h) = h^2, c(h) = n-1+h, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 6, 2, 20, 11, 3, 50, 34, 16, 4, 105, 80, 48, 21, 5, 196, 160, 110, 62, 26, 6, 336, 287, 215, 140, 76, 31, 7, 540, 476, 378, 270, 170, 90, 36, 8, 825, 744, 616, 469, 325, 200, 104, 41, 9, 1210, 1110, 948, 756, 560, 380, 230, 118, 46, 10

Offset: 1

Clark Kimberling, Jun 16 2012

Principal diagonal:  A117066
Antidiagonal sums:  A033455
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....6....20....50....105....196...336
2....11...34....80....160....287...476
3....16...48....110...215....378...616
4....21...62....140...270....469...756
5....26...76....170...325....560...896
...
T(5,1) = (1)**(5) = 5
T(5,2) = (1,4)**(5,6) = 1*6+4*5 = 26
T(5,3) = (1,4,9)**(5,6,7) = 1*7+4*6+9*5 = 76

G. C. Greubel, Antidiagonal rows n = 1..100

Cf. A213500.

Flat(List([1..12], n-> List([1..n], k-> (n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12))); # G. C. Greubel, Jul 05 2019

[[(n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 05 2019

(* First program *)
b[n_]:= n^2; 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}]] (* A213503 *)
r[n_]:= Table[T[n, k], {k,40}]  (* columns of antidiagonal triangle *)
d = Table[T[n, n], {n, 1, 40}] (* A117066 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A033455 *)
(* Second program *)
Table[(n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 05 2019 *)

t(n,k) = (n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12;
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 05 2019

[[(n-k+1)*((n-k+1)^3 + 4*(n-k+1)^2*k + 6*k*(n-k+1) - n + 3*k - 1)/12 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 05 2019

T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).
G.f. for row n:  f(x)/g(x), where f(x) = n + x - (n - 1)^2 x^2 and g(x) = (1 - x)^5.
T(n,k) = k*(k^3 + 4*k^2*n + 6*k*n - k + 2*n)/12. - G. C. Greubel, Jul 05 2019

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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A185874 Second accumulation array of A051340, by antidiagonals.

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A321500 Triangular table T(n,k) = (n+k)*(n^2+k^2), n >= k >= 0; read by rows n = 0, 1, 2, ...

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A213503 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

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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.

A213503 Rectangular array: (row n) = bc, where b(h) = h^2, c(h) = n-1+h, n>=1, h>=1, and = convolution.