A213577 -id:A213577 - 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.

A213576 Rectangular array: (row n) = bc, where b(h) = h, c(h) = F(n-1+h), where F=A000045 (Fibonacci numbers), n >= 1, h >= 1, and = convolution.

Original entry on oeis.org

1, 3, 1, 7, 4, 2, 14, 10, 7, 3, 26, 21, 17, 11, 5, 46, 40, 35, 27, 18, 8, 79, 72, 66, 56, 44, 29, 13, 133, 125, 118, 106, 91, 71, 47, 21, 221, 212, 204, 190, 172, 147, 115, 76, 34, 364, 354, 345, 329, 308, 278, 238, 186, 123, 55, 596, 585, 575, 557, 533, 498, 450, 385, 301, 199, 89

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal:  A213577.
Antidiagonal sums:  A213578.
Row 1,  (1,2,3,...)**(1,1,2,3,5,...): A001924;
Row 2,  (1,2,3,...)**(1,2,3,5,8,...): A001891;
Row 3,  (1,2,3,...)**(2,3,5,8,13,...): A033937;
Row 4,  (1,2,3,...)**(3,5,8,13,21,...): A033960;
Row 5,  (1,2,3,...)**(5,8,13,21,...): A037140;
Row 6,  (1,2,3,...)**(8,13,21,34,...): A037157.
For a guide to related arrays, see A213500.
The falling antidiagonal rows can be computed by the sum Sum_{j=0..n-k} (n-k-j+1)*Fibonacci(k+j) which can also be seen as Fibonacci(n+4) - Lucas(k+2) - (n-k)*Fibonacci(k+1). - G. C. Greubel, Jul 05 2019

			Northwest corner (the array is read by falling antidiagonals):
  1,   3,   7,  14,  26,  46,  79
  1,   4,  10,  21,  40,  72, 125
  2,   7,  17,  35,  66, 118, 204
  3,  11,  27,  56, 106, 190, 329
  5,  18,  44,  91, 172, 308, 533
  8,  29,  71, 147, 278, 498, 862

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

Cf. A213500.

Flat(List([1..10], n-> List([1..n], k-> Fibonacci(n+4) - (n-k+1) *Fibonacci(k+1) - Fibonacci(k+3)))); # G. C. Greubel, Jul 05 2019

[[Fibonacci(n+4) -(n-k)*Fibonacci(k+1) -Lucas(k+2): k in [1..n]]: n in [1..10]]; // G. C. Greubel, Jul 05 2019

(* First Program *)
b[n_]:= n; c[n_]:= Fibonacci[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}]] (* A213576 *)
r[n_]:= Table[t[n, k], {k,1,40}]  (* columns of antidiagonal triangle *)
d = Table[t[n, n], {n, 1, 40}] (* A213577 *)
s[n_]:= Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213578 *)
(* Second Program *)
T[n_, k_]:= Fibonacci[n+4] - (n-k)*Fibonacci[k+1] - LucasL[k+2];
Table[T[n,k], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Jul 05 2019 *)

T(n, k)= fibonacci(n+4) - (n-k+1)*fibonacci(k+1) - fibonacci(k+3);
for(n=1,10, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 05 2019

[[fibonacci(n+4) - (n-k+1)*fibonacci(k+1) - fibonacci(k+3) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Jul 05 2019

Rows: T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).
Columns: T(n,k) = T(n-1,k) + T(n-2,k).
G.f. for row n: f(x)/g(x), where f(x) = F(n) - F(n-1)*x and g(x) = (1 - x - x^2)*(1 - x)^2.
T(n,k) = F(n+k+3) - k*F(n+1) - F(n+3). - Ehren Metcalfe, Jul 04 2019

A213578 Antidiagonal sums of the convolution array A213576.

Original entry on oeis.org

1, 4, 13, 34, 80, 174, 359, 712, 1371, 2580, 4768, 8684, 15629, 27852, 49225, 86390, 150704, 261530, 451795, 777360, 1332791, 2277864, 3882048, 6599064, 11191705, 18940564, 31992709, 53943562, 90807056, 152631750, 256190783

Offset: 1

Clark Kimberling, Jun 18 2012

Clark Kimberling, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,4,0,-1).

Cf. A213576, A213500.

List([1..40], n-> n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5)); # G. C. Greubel, Jul 05 2019

[n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5): n in [1..40]]; // Vincenzo Librandi, Jul 05 2019

b[n_]:= n; c[n_]:= Fibonacci[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}]] (* A213576 *)
r[n_] := Table[t[n, k], {k,40}]  (* columns of antidiagonal triangle *)
d = Table[t[n, n], {n,1,40}] (* A213577 *)
s[n_] := Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213578 *)
(* alternate program *)
LinearRecurrence[{4,-4,-2,4,0,-1},{1,4,13,34,80,174},40] (* Harvey P. Dale, Jul 04 2019 *)

vector(40, n, n*fibonacci(n+4)-2*(fibonacci(n+5)-n-5)) \\ G. C. Greubel, Jul 05 2019

[n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5) for n in (1..40)] # G. C. Greubel, Jul 05 2019

a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) - a(n-6).
G.f.: (1 + x^2)/(1 - 2*x + x^3)^2.
a(n) = n*F(n+4) - 2*(F(n+5) - n - 5), F = A000045. - Ehren Metcalfe, 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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A213576 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = F(n-1+h), where F=A000045 (Fibonacci numbers), n >= 1, h >= 1, and ** = convolution.

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A213578 Antidiagonal sums of the convolution array A213576.

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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.

A213576 Rectangular array: (row n) = bc, where b(h) = h, c(h) = F(n-1+h), where F=A000045 (Fibonacci numbers), n >= 1, h >= 1, and = convolution.