A213587 -id:A213587 - cp's OEIS frontend

A213588 Principal diagonal of the convolution array A213587.

1, 7, 27, 96, 315, 994, 3043, 9123, 26909, 78370, 225911, 645732, 1832677, 5170111, 14509695, 40537284, 112805043, 312808198, 864707719, 2383649115, 6554153921, 17980221382, 49222822127, 134495771976, 366850762825

Offset: 1

Clark Kimberling, Jun 19 2012

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-5,-5,5,-1).

Cf. A213587, A213500.

List([1..30], n-> (n*Lucas(1,-1,2*n+2)[2] - Fibonacci(n)*Lucas(1,-1,n-1)[2])/5); # G. C. Greubel, Jul 08 2019

[(n*Lucas(2*n+2) - Fibonacci(n)*Lucas(n-1))/5: n in [1..30]]; // G. C. Greubel, Jul 08 2019

(* First program *)
b[n_]:= Fibonacci[n+1]; c[n_]:= Fibonacci[n+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}]] (* A213587 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213588 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213589 *)
(* Second program *)
Table[(n*LucasL[2n+2] -Fibonacci[n]*LucasL[n-1])/5, {n, 30}] (* G. C. Greubel, Jul 08 2019 *)

lucas(n) = fibonacci(n+1) + fibonacci(n-1);
vector(30, n, (n*lucas(2*n+2) - fibonacci(n)*lucas(n-1))/5) \\ G. C. Greubel, Jul 08 2019

[(n*lucas_number2(2*n+2,1,-1) - fibonacci(n)*lucas_number2(n-1, 1, -1))/5 for n in (1..30)] # G. C. Greubel, Jul 08 2019

a(n) = 5*a(n-1) - 5*a(n-2) - 5*a(n-3) + 5*a(n-4) - a(n-5).
G.f.: x*(1 + 2*x - 3*x^2 + x^3)/((1 + x)*(1 - 3*x + x^2)^2).
a(n) = (n*Lucas(2*n+2) - Fibonacci(n)*Lucas(n-1))/5. - G. C. Greubel, Jul 08 2019

A213589 Antidiagonal sums of the convolution array A213587.

Original entry on oeis.org

1, 6, 20, 55, 135, 308, 668, 1395, 2830, 5610, 10914, 20904, 39515, 73860, 136720, 250937, 457137, 827260, 1488190, 2662905, 4741946, 8407236, 14846100, 26120400, 45801925, 80064018, 139553708, 242597035, 420678315, 727792580

Offset: 1

Clark Kimberling, Jun 19 2012

Clark Kimberling, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. A213587, A213500.

F:=Fibonacci;; List([1..35], n-> (n+1)*((n+2)*F(n+3) + 2*(n-2)*F(n+2))/10) # G. C. Greubel, Jul 08 2019

F:=Fibonacci; [(n+1)*((n+2)*F(n+3) + 2*(n-2)*F(n+2))/10: n in [1..35]]; // G. C. Greubel, Jul 08 2019

(* First program *)
b[n_]:= Fibonacci[n+1]; c[n_]:= Fibonacci[n+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}]] (* A213587 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213588 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213589 *)
(* Second program *)
Table[(n+1)*(n*LucasL[n+3] -2*Fibonacci[n])/10, {n, 35}] (* G. C. Greubel, Jul 08 2019 *)

a(n):=(n+1)/2*sum((n-j)*binomial(n-j+1,j),j,0,(n+1)/2); /* Vladimir Kruchinin, Apr 09 2016 */

vector(35, n, f=fibonacci; (n+1)*((n+2)*f(n+3)+ 2*(n-2)*f(n+2) )/10) \\ G. C. Greubel, Jul 08 2019

f=fibonacci; [(n+1)*((n+2)*f(n+3)+ 2*(n-2)*f(n+2) )/10 for n in (1..35)] # G. C. Greubel, Jul 08 2019

a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6).
G.f.: x*(1 + 3*x + 2*x^2)/(1 - x - x^2)^3.
a(n) = (n+1)/2*Sum_{j=0..(n+1)/2}((n-j)*binomial(n-j+1,j)). - Vladimir Kruchinin, Apr 09 2016
a(n) = (n+1)*(n*Lucas(n+3) - 2*Fibonacci(n))/10 = (n+1)*((n+2) *Fibonacci(n+3) + 2*(n-2)*Fibonacci(n+2))/10. - G. C. Greubel, Jul 08 2019

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

Original entry on oeis.org

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.

A213590 Rectangular array: (row n) = bc, where b(h) = h^2, c(h) = F(n-1+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 5, 1, 15, 6, 2, 36, 20, 11, 3, 76, 51, 35, 17, 5, 148, 112, 87, 55, 28, 8, 273, 224, 188, 138, 90, 45, 13, 485, 421, 372, 300, 225, 145, 73, 21, 839, 758, 694, 596, 488, 363, 235, 118, 34, 1424, 1324, 1243, 1115, 968, 788, 588, 380, 191, 55, 2384, 2263, 2163, 2001, 1809, 1564, 1276, 951, 615, 309, 89

Offset: 1

Clark Kimberling, Jun 19 2012

Principal diagonal: A213504.
Antidiagonal sums: A213557.
Row 1,  (1,4,9,16,...)**(1,1,2,3,5,...): A053808.
Row 2,  (1,4,9,16,...)**(1,2,3,5,8,...): A213586.
Row 3,  (1,4,9,16,...)**(2,3,5,8,13,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....5....15....36....76.....148
1....6....20....51....112....224
2....11...35....87....188....372
3....17...55....138...300....596
5....28...90....225...488....868
8....45...145...363...788....1564
13...73...235...588...1276...2532

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

Cf. A213500, A213587.

F:=Fibonacci;; Flat(List([1..12],n-> List([1..n],k-> F(n+7)-F(k+6) -2*(n-k+1)*F(k+3)-(n-k+1)^2*F(k+1) ))) # G. C. Greubel, Jul 05 2019

F:=Fibonacci; [[F(n+7) -F(k+6) -2*(n-k+1)*F(k+3) -(n-k+1)^2 *F(k+1): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 05 2019

(* First program *)
b[n_]:= n^2; 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}]] (* A213590 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213504 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213557 *)
(* Second program *)
t[n_, k_]:= Fibonacci[n+7] - Fibonacci[k+6] - 2*(n-k+1)*Fibonacci[k+3] - (n-k+1)^2*Fibonacci[k+1]; Table[t[n, k], {n, 1, 12}, {k, 1, n}]//Flatten (* G. C. Greubel, Jul 05 2019 *)

f=fibonacci; t(n,k) = f(n+7) -f(k+6) -2*(n-k+1)*f(k+3) -(n-k+1)^2 *f(k+1);
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 05 2019

f=fibonacci; [[f(n+7) -f(k+6) -2*(n-k+1)*f(k+3) - (n-k+1)^2* f(k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 05 2019

Rows:  T(n,k) = 4*T(n,k-1) -5*T(n,k-2) +*T(n,k-3) +2*T(n,k-4) -T(n,k-5).
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 + F(n-1)*x^2 and g(x) = (1 - x - x^2)*(1 - x )^3.
T(n, k) = Fibonacci(n+k+6) - Fibonacci(n+6) - 2*k*Fibonacci(n+3) - k^2*Fibonacci(n+1). - G. C. Greubel, Jul 05 2019

A213571 Rectangular array: (row n) = bc, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 5, 3, 16, 13, 7, 42, 38, 29, 15, 99, 94, 82, 61, 31, 219, 213, 198, 170, 125, 63, 466, 459, 441, 406, 346, 253, 127, 968, 960, 939, 897, 822, 698, 509, 255, 1981, 1972, 1948, 1899, 1809, 1654, 1402, 1021, 511, 4017, 4007, 3980, 3924, 3819, 3633

Offset: 1

Clark Kimberling, Jun 19 2012

Principal diagonal:  A213572.
Antidiagonal sums:  A213581.
Row 1,  (1,2,3,4,5,...)**(1,3,7,15,31,...): A002662.
Row 2,  (1,2,3,4,5,...)**(3,7,15,31,63,...).
Row 3,  (1,2,3,4,5,...)**(7,15,31,63,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
   1,    5,   16,   42,   99,  219, ...
   3,   13,   38,   94,  213,  459, ...
   7,   29,   82,  198,  441,  939, ...
  15,   61,  170,  406,  897, 1899, ...
  31,  125,  346,  822, 1809, 3819, ...
  ...

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

Cf. A213500, A213587.

Flat(List([1..12], n-> List([1..n], k-> 2^(n+2) -2^k*(n-k+3) -Binomial(n-k+2, 2) ))); # G. C. Greubel, Jul 25 2019

[2^(n+2) -2^k*(n-k+3) -Binomial(n-k+2, 2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 25 2019

(* First program *)
b[n_]:= n; c[n_]:= -1 + 2^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}]  (* A213571 *)
d = Table[t[n, n], {n, 1, 40}] (* A213572 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213581 *)
(* Additional programs *)
Table[2^(n+2) -2^k*(n-k+3) -Binomial[n-k+2, 2], {n,12}, {k, n}]//Flatten (* G. C. Greubel, Jul 25 2019 *)

for(n=1,12, for(k=1,n, print1(2^(n+2) -2^k*(n-k+3) -binomial(n-k+2, 2), ", "))) \\ G. C. Greubel, Jul 25 2019

[[2^(n+2) -2^k*(n-k+3) -binomial(n-k+2, 2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 25 2019

T(n,k) = 5*T(n,k-1) - 9*T(n,k-2) + 7*T(n,k-3) - 2*T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(-1 + 2^n - (-2 + 2^n)*x) and g(x) = (1 - 2*x)(1 - x)^3.
T(n,k) = 2^(n+k+1) - 2^n*(k+2) - binomial(k+1, 2). - G. C. Greubel, Jul 25 2019

A213566 Rectangular array: (row n) = bc, where b(h) = F(h), c(h) = (n-1+h)^2, F = A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 5, 4, 15, 13, 9, 36, 33, 25, 16, 76, 71, 59, 41, 25, 148, 140, 120, 93, 61, 36, 273, 260, 228, 183, 135, 85, 49, 485, 464, 412, 340, 260, 185, 113, 64, 839, 805, 721, 604, 476, 351, 243, 145, 81, 1424, 1369, 1233, 1044, 836, 636, 456, 309, 181, 100

Offset: 1

Clark Kimberling, Jun 19 2012

Principal diagonal:  A213567.
Antidiagonal sums:  A213570.
Row 1,  (1,1,2,3,5,...)**(1,4,9,16,25,...): A053808.
Row 2,  (1,1,2,3,5,...)**(4,9,16,25,...).
Row 3,  (1,1,2,3,5,...)**(16,25,49,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....5....15....36....76
4....13...33....71....140
9....25...59....120...228
16...41...93....183...340
25...61...135...260...476

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

Cf. A213500, A213587.

F:=Fibonacci;; Flat(List([1..12], n-> List([1..n], k-> k*(k*F(n-k+3) +2*F(n-k+4)) + F(n-k+7) -(k+2)*(2*n-k+4) -(n-k+1)^2 -4 ))); # G. C. Greubel, Jul 26 2019

F:=Fibonacci; [k*(k*F(n-k+3) +2*F(n-k+4)) + F(n-k+7) -(k+2)*(2*n-k+4) -(n-k+1)^2 -4: k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 26 2019

(* First program *)
b[n_]:= Fibonacci[n]; c[n_]:= n^2;
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}]  (* A213566 *)
d = Table[t[n, n], {n, 1, 40}] (* A213567 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213570 *)
(* Second program *)
With[{F = Fibonacci}, Table[k*(k*F[n-k+3] +2*F[n-k+4]) + F[n-k+7] -(k+2) *(2*n-k+4) -(n-k+1)^2 -4, {n, 12}, {k, n}]//Flatten] (* G. C. Greubel, Jul 26 2019 *)

f=fibonacci;
for(n=1,12, for(k=1,n, print1(k*(k*f(n-k+3) +2*f(n-k+4)) + f(n-k+7) -(k+2)*(2*n-k+4) -(n-k+1)^2 -4, ", "))) \\ G. C. Greubel, Jul 26 2019

f=fibonacci; [[k*(k*f(n-k+3) +2*f(n-k+4)) + f(n-k+7) -(k+2)*(2*n-k+4) -(n-k+1)^2 -4 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 26 2019

T(n,k) = 4*T(n,k-1)-5*T(n,k-2)+T(n,k-3)+2*T(n,k-4)-T(n,k-5).
G.f. for row n:  f(x)/g(x), where f(x) = x*(n^2 - (2*n^2 - 2*n - 1)*x + (n - 1)^2 *x^2) and g(x) = (1 - x - x^2)*(1 - x )^3.
T(n,k) = n*(n*F(k+2) + 2*F(k+3)) + F(k+6) - (n+2)*(2*k+n+2) - k^2 - 4, F = A000045. - Ehren Metcalfe, Jul 10 2019

cp's OEIS Frontend

A213588 Principal diagonal of the convolution array A213587.

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A213589 Antidiagonal sums of the convolution array A213587.

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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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A213590 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = F(n-1+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

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A213571 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

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A213566 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = (n-1+h)^2, F = A000045 (Fibonacci numbers), 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.

A213590 Rectangular array: (row n) = bc, where b(h) = h^2, c(h) = F(n-1+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

A213571 Rectangular array: (row n) = bc, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and = convolution.

A213566 Rectangular array: (row n) = bc, where b(h) = F(h), c(h) = (n-1+h)^2, F = A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.