A132117 -id:A132117 - 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.

A143328 Table T(n,k) read by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary Lyndon words (n,k >= 1) with length less than or equal to n.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 5, 1, 5, 10, 14, 8, 1, 6, 15, 30, 32, 14, 1, 7, 21, 55, 90, 80, 23, 1, 8, 28, 91, 205, 294, 196, 41, 1, 9, 36, 140, 406, 829, 964, 508, 71, 1, 10, 45, 204, 728, 1960, 3409, 3304, 1318, 127, 1, 11, 55, 285, 1212, 4088, 9695, 14569, 11464, 3502, 226, 1

Offset: 1

Alois P. Heinz, Aug 07 2008

			T(3,2) = 5, because 5 words of length <=3 over 2-letter alphabet {a,b} are primitive Lyndon words: a, b, ab, aab, abb.
Table begins:
  1,  2,  3,   4,   5,  ...
  1,  3,  6,  10,  15,  ...
  1,  5, 14,  30,  55,  ...
  1,  8, 32,  90, 205,  ...
  1, 14, 80, 294, 829,  ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index entries for sequences related to Lyndon words

Columns k=1-5 give: A000012, A062692, A114945, A114946, A114947.
Rows n=1-4 give: A000027, A000217, A000330, A132117.
Main diagonal gives A215475.
Cf. A143324, A074650, A008683.

with(numtheory):
f0:= proc(n) option remember; unapply(k^n-add(f0(d)(k),
        d=divisors(n)minus{n}), k)
     end:
f2:= proc(n) option remember; unapply(f0(n)(x)/n, x) end:
g2:= proc(n) option remember; unapply(add(f2(j)(x), j=1..n), x) end:
T:= (n,k)-> g2(n)(k):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

f0[n_] := f0[n] = Function[k, k^n-Sum[f0[d][k], {d, Divisors[n]//Most}]]; f2[n_] := f2[n] = Function[x, f0[n][x]/n]; g2[n_] := g2[n] = Function[x, Sum[f2[j][x], {j, 1, n}]]; T[n_, k_] := g2[n][k]; Table[T[n, 1+d-n], {d, 1, 12}, {n, 1, d}]//Flatten (* Jean-François Alcover, Feb 12 2014, translated from Maple *)

T(n,k) = Sum_{1<=j<=n} (1/j) * Sum_{d|j} mu(j/d)*k^d.

T(n,k) = Sum_{1<=j<=n} A074650(j,k).

A213771 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 2, 18, 11, 3, 40, 30, 16, 4, 75, 62, 42, 21, 5, 126, 110, 84, 54, 26, 6, 196, 177, 145, 106, 66, 31, 7, 288, 266, 228, 180, 128, 78, 36, 8, 405, 380, 336, 279, 215, 150, 90, 41, 9, 550, 522, 472, 406, 330, 250, 172, 102, 46

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213772
Antidiagonal sums: A132117
Row 1, (1,4,7,10,...)**(1,2,3,4,...): A002411
Row 2, (1,4,7,10,...)**(2,3,4,5,...): A162260
Row 3, (1,2,3,4,5,...)**(7,10,13,16,...): (k^3 + 7*k^2 - 2*k)/2
Row 4, (1,2,3,4,5,...)**(10,13,16,...): (k^3 + 10*k^2 - 3*k)/2
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1....6....18...40....75....126
2....11...30...62....110...177
3....16...42...84....145...228
4....21...54...106...180...279
5....26...66...128...215...330

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

Cf. A212500

b[n_]:=3n-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}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213771 *)
Table[t[n,n],{n,1,40}] (* A213772 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A132117 *)

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*(n + (n+1)*x - (n+2)*x^2) and g(x) = (1 - x)^4.

A176060 a(n) = n(n+1)(3n^2+5n+4)/12.

Original entry on oeis.org

0, 2, 13, 46, 120, 260, 497, 868, 1416, 2190, 3245, 4642, 6448, 8736, 11585, 15080, 19312, 24378, 30381, 37430, 45640, 55132, 66033, 78476, 92600, 108550, 126477, 146538, 168896, 193720, 221185, 251472, 284768, 321266, 361165, 404670, 451992

Offset: 0

Bruno Berselli, Dec 06 2010

Antidiagonal sums of the convolution array A213821. [Clark Kimberling, Jul 04 2012]

			For n=5, a(5)=1*(1*0+6)+2*(2*1+6)+3*(3*2+6)+4*(4*3+6)+5*(5*4+6)=260.

"Supplemento al Periodico di Matematica", Raffaello Giusti Editore (Livorno), May 1908, p. 111 (Problem 923).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A213821.

[n*(n+1)*(3*n^2+5*n+4)/12: n in [0..40]]; // Vincenzo Librandi, Jul 02 2011

Table[n(n+1)(3n^2+5n+4)/12,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,2,13,46,120},40] (* Harvey P. Dale, Jul 14 2011 *)

G.f.: x*(2+3*x+x^2)/(1-x)^5.
a(n) = sum(k*(k*(k-1)+n+1), k=1..n) with n>0 (summation proposed in the Problem 923, see References).
a(0)=0, a(1)=2, a(2)=13, a(3)=46, a(4)=120, a(n)=5*a(n-1)- 10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Jul 14 2011
a(-n) = A132117(n-1) with A132117(-1)=A132117(0)=0. - Bruno Berselli, Aug 22 2011

A178067 Triangle read by rows: T(n,k) = (n^2 + k)*(n - k + 1)/2.

Original entry on oeis.org

1, 5, 3, 15, 11, 6, 34, 27, 19, 10, 65, 54, 42, 29, 15, 111, 95, 78, 60, 41, 21, 175, 153, 130, 106, 81, 55, 28, 260, 231, 201, 170, 138, 105, 71, 36, 369, 332, 294, 255, 215, 174, 132, 89, 45, 505, 459, 412, 364, 315, 265, 214, 162, 109, 55, 671, 615, 558, 500, 441, 381, 320, 258, 195, 131, 66

Offset: 1

Gary W. Adamson, May 18 2010

			First few rows of the triangle:
    1;
    5,   3;
   15,  11,   6;
   34,  27,  19,  10;
   65,  54,  42,  29,  15;
  111,  95,  78,  60,  41,  21;
  175, 153, 130, 106,  81,  55,  28;
  260, 231, 201, 170, 138, 105,  71,  36;
  369, 332, 294, 255, 215, 174, 132,  89,  45;
  505, 459, 412, 364, 315, 265, 214, 162, 109,  55;
  671, 615, 558, 500, 441, 381, 320, 258, 195, 131, 66;
  ...
Example: Row 3 = (15, 11 6) since row 3 of A000027 as a triangle = (4, 5, 6); then taking partial sums from the right.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Column 1 is A006003.
Row sums are A132117.
Main diagonal is A000217.

As infinite lower triangular matrices, A000027 * A000012 (the all 1's triangle).

Name changed and terms a(37) and beyond from Andrew Howroyd, Apr 17 2021

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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A143328 Table T(n,k) read by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary Lyndon words (n,k >= 1) with length less than or equal to n.

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A213771 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

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A176060 a(n) = n*(n+1)*(3*n^2+5*n+4)/12.

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A178067 Triangle read by rows: T(n,k) = (n^2 + k)*(n - k + 1)/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.

A176060 a(n) = n(n+1)(3n^2+5n+4)/12.