A213579 -id:A213579 - cp's OEIS frontend

A213580 Principal diagonal of the convolution array A213579.

1, 5, 15, 35, 74, 146, 277, 511, 925, 1651, 2916, 5108, 8889, 15385, 26507, 45491, 77806, 132678, 225645, 382835, 648121, 1095075, 1846920, 3109800, 5228209, 8777261, 14716167, 24643331, 41220050, 68873786, 114964741, 191719783

Offset: 1

Clark Kimberling, Jun 18 2012

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,4,0,-1).

Cf. A000045, A213500, A213579.

F:=Fibonacci;; List([1..40], n-> F(n+3) +n*F(n+2) -2*(n+1)); # G. C. Greubel, Jul 08 2019

F:=Fibonacci; [F(n+3) + n*F(n+2) -2*(n+1): n in [1..40]]; // G. C. Greubel, Jul 08 2019

(* First program *)
b[n_]:= Fibonacci[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}]] (* A213579 *)
r[n_]:= Table[T[n, k], {k, 40}]
d = Table[T[n, n], {n, 1, 40}] (* A213580 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A053808 *)
(* Second program *)
Table[Fibonacci[n+3] + n*Fibonacci[n+2] -2*(n+1), {n, 40}] (* G. C. Greubel, Jul 08 2019 *)

vector(40, n, f=fibonacci; f(n+3) +n*f(n+2) -2*(n+1)) \\ G. C. Greubel, Jul 08 2019

f=fibonacci; [f(n+3) +n*f(n+2) -2*(n+1) for n in (1..40)] # G. C. Greubel, Jul 08 2019

a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) + a(n-5).
G.f.: x*(1 + x - x^2 - 3*x^3)/(1 - 2*x + x^3)^2.
a(n) = Fibonacci(n+3) + n*Fibonacci(n+2) - 2*(n+1). - 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.

A053808 Partial sums of A001891.

Original entry on oeis.org

1, 5, 15, 36, 76, 148, 273, 485, 839, 1424, 2384, 3952, 6505, 10653, 17383, 28292, 45964, 74580, 120905, 195885, 317231, 513600, 831360, 1345536, 2177521, 3523733, 5701983, 9226500, 14929324, 24156724, 39087009, 63244757, 102332855, 165578768, 267912848

Offset: 0

Barry E. Williams, Mar 27 2000

Antidiagonal sums of the convolution array A213579 and row 1 of the convolution array A213590. - Clark Kimberling, Jun 18 2012

Also number CG(n,2) of complete games with n players of 2 types. - N. J. A. Sloane, Dec 29 2012

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. Freixas and S. Kurz, The golden number and Fibonacci sequences in the design of voting structures, 2012. - From _N. J. A. Sloane_, Dec 29 2012
W. Lang, Problem B-858, Fibonacci Quarterly, 36,3 (1998) 373-374; Solution, ibid. 37,2 (1999) 183-184.
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Convolution of A000290 (squares) with A000045, n >= 1. (Fibonacci) - Wolfdieter Lang, Apr 10 2000

Right-hand column 7 of triangle A011794.

List([0..40], n-> Fibonacci(n+8) - (n^2 +8*n+20)); # G. C. Greubel, Jul 06 2019

[Fibonacci(n+8) - (n^2+8*n+20): n in [0..40]]; // G. C. Greubel, Jul 06 2019

Table[Fibonacci[n+8] -(n^2 +8*n+20), {n,0,40}] (* G. C. Greubel, Jul 06 2019 *)
LinearRecurrence[{4,-5,1,2,-1},{1,5,15,36,76},40] (* Harvey P. Dale, Apr 14 2022 *)

vector(40, n, n--; fibonacci(n+8) - (n^2 +8*n+20)) \\ G. C. Greubel, Jul 06 2019

[fibonacci(n+8) - (n^2 +8*n+20) for n in (0..20)] # G. C. Greubel, Jul 06 2019

a(n) = a(n-1) + a(n-2) + (n+1)^2, a(-n)=0.
G.f.: (1+x)/((1-x-x^2)*(1-x)^3).
a(n) = Fibonacci(n+6) - (n^2 + 4*n + 8), n >= 2 (see p. 184 of FQ reference).
a(n-2) = Sum_{i=0..n} Fibonacci(i)*(n-i)^2. - Benoit Cloitre, Mar 06 2004

A213584 Rectangular array: (row n) = bc, where b(h) = F(h+1), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 4, 2, 10, 7, 3, 21, 16, 10, 4, 40, 32, 22, 13, 5, 72, 59, 43, 28, 16, 6, 125, 104, 78, 54, 34, 19, 7, 212, 178, 136, 97, 65, 40, 22, 8, 354, 299, 231, 168, 116, 76, 46, 25, 9, 585, 496, 386, 284, 200, 135, 87, 52, 28, 10, 960, 816, 638, 473, 337, 232, 154, 98, 58, 31, 11

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal: A213585.
Antidiagonal sums: A213586.
Row 1, (1,2,3,5,...)**(1,2,3,4,...): A001891.
Row 2, (1,2,3,5,...)**(2,3,4,5,...): A023550.
Row 3, (1,2,3,5,...)**(3,4,5,6,...): A023554.
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...4....10...21...40....72
2...7....16...32...59....104
3...10...22...43...78....136
4...13...28...54...97....168
5...16...34...65...116...200
6...19...40...76...135...232

Clark Kimberling, Antidiagonals n = 1..60

Cf. A000045, A213500, A213579.

Flat(List([1..12], n-> List([1..n], k-> Fibonacci(n-k+5) + k*Fibonacci(n-k+4) -(2*n+5)))) # G. C. Greubel, Jul 08 2019

[[Fibonacci(n-k+5) + k*Fibonacci(n-k+4) -(2*n+5): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019

(* First program *)
b[n_]:= Fibonacci[n+1]; 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}]] (* A213584 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
d = Table[T[n, n], {n, 1, 40}] (* A213585 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213586 *)
(* Second program *)
Table[Fibonacci[n-k+5] + k*Fibonacci[n-k+4] -2*n-5, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)

t(n,k) = fibonacci(n-k+5) + k*fibonacci(n-k+4) -(2*n+5);
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019

[[fibonacci(n-k+5) + k*fibonacci(n-k+4) -(2*n+5) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019

T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).
G.f. for row n:  f(x)/g(x), where f(x) = n + x - (n - 1)*x and g(x) = (1 - x - x^2)*(1 - x)^2.
T(n, k) = Fibonacci(k+4) + n*Fibonacci(k+3) - 2*(n+k) - 3. - G. C. Greubel, Jul 08 2019

A023552 Convolution of natural numbers >= 3 and Fibonacci numbers.

Original entry on oeis.org

3, 7, 15, 28, 50, 86, 145, 241, 397, 650, 1060, 1724, 2799, 4539, 7355, 11912, 19286, 31218, 50525, 81765, 132313, 214102, 346440, 560568, 907035, 1467631, 2374695, 3842356, 6217082, 10059470, 16276585, 26336089, 42612709, 68948834, 111561580, 180510452

Offset: 1

Clark Kimberling

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000032, A000045, A213579.

F:=Fibonacci; List([1..40], n-> F(n+4)+2*F(n+2)-n-5); # G. C. Greubel, Jul 08 2019

F:=Fibonacci; [F(n+4)+2*F(n+2)-n-5: n in [1..40]]; // G. C. Greubel, Jul 08 2019

LinearRecurrence[{3,-2,-1,1},{3,7,15,28},40] (* or *) Rest[ CoefficientList[Series[(x(3-2x))/((1-x-x^2)(1-x)^2),{x,0,40}],x]]  (* Harvey P. Dale, Apr 24 2011 *)
With[{F=Fibonacci}, Table[F[n+4]+2*F[n+2]-n-5, {n,40}]] (* G. C. Greubel, Jul 08 2019 *)

Vec(x*(3-2*x)/((1-x-x^2)*(1-x)^2) + O(x^40)) \\ Colin Barker, Mar 11 2017

vector(40, n, f=fibonacci; f(n+4)+2*f(n+2)-n-5) \\ G. C. Greubel, Jul 08 2019

f=fibonacci; [f(n+4)+2*f(n+2)-n-5 for n in (1..40)] # G. C. Greubel, Jul 08 2019

G.f.: x*(3-2*x)/((1-x-x^2)*(1-x)^2). - Ralf Stephan, Apr 28 2004
From Colin Barker, Mar 11 2017: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4.
a(n) = -5 + (2^(-1-n)*((1-sqrt(5))^n*(-13+5*sqrt(5)) + (1+sqrt(5))^n*(13+5*sqrt(5)))) / sqrt(5) - n. (End)
a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (n+5). - G. C. Greubel, Jul 08 2019

A210730 a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=a(1)=0.

Original entry on oeis.org

0, 0, 4, 9, 19, 35, 62, 106, 178, 295, 485, 793, 1292, 2100, 3408, 5525, 8951, 14495, 23466, 37982, 61470, 99475, 160969, 260469, 421464, 681960, 1103452, 1785441, 2888923, 4674395, 7563350, 12237778, 19801162, 32038975, 51840173, 83879185, 135719396

Offset: 0

Alex Ratushnyak, May 10 2012

Deleting the 0's leaves row 4 of the convolution array A213579. - Clark Kimberling, Jun 20 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A033818: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=0 (except first 2 terms and sign).
Cf. A002062: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=0 (except the first term and sign).
Cf. A065220: a(n)=a(n-1)+a(n-2)+n-3, a(0)=a(1)=0.
Cf. A001924: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=0 (except the first term).
Cf. A023548: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=0 (except first 2 terms).
Cf. A023552: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=0 (except first 2 terms).
Cf. A210731: a(n)=a(n-1)+a(n-2)+n+3, a(0)=a(1)=0.
Cf. A000045, A213579.

F:=Fibonacci;; List([0..40], n-> F(n+3)+3*F(n+1)-n-5); # G. C. Greubel, Jul 08 2019

I:=[0, 0, 4, 9]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-Self(n-3)+Self(n-4): n in [1..37]]; // Bruno Berselli, May 10 2012

F:=Fibonacci; [F(n+3)+3*F(n+1)-n-5: n in [0..40]]; // G. C. Greubel, Jul 08 2019

RecurrenceTable[{a[0]==a[1]==0, a[n]==a[n-1] +a[n-2] +n+2}, a, {n, 40}] (* Bruno Berselli, May 10 2012 *)
LinearRecurrence[{3,-2,-1,1},{0,0,4,9},40] (* Harvey P. Dale, Jul 24 2013 *)
With[{F=Fibonacci}, Table[F[n+3]+2*F[n+1]-n-5, {n, 40}]] (* G. C. Greubel, Jul 08 2019 *)

concat(vector(2), Vec(x^2*(4-3*x)/((1-x)^2*(1-x-x^2)) + O(x^50))) \\ Colin Barker, Mar 11 2017

vector(40, n, n--; f=fibonacci; f(n+3)+3*f(n+1)-n-5) \\ G. C. Greubel, Jul 08 2019

f=fibonacci; [f(n+3)+3*f(n+1)-n-5 for n in (0..40)] # G. C. Greubel, Jul 08 2019

G.f.: x^2*(4-3*x)/((1-x)^2*(1-x-x^2)). - Bruno Berselli, May 10 2012
a(n) = A210677(n)-1. - Bruno Berselli, May 10 2012
a(0)=0, a(1)=0, a(2)=4, a(3)=9, a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). - Harvey P. Dale, Jul 24 2013
a(n) = -5 + (2^(-1-n)*((1-sqrt(5))^n*(-7+5*sqrt(5)) + (1+sqrt(5))^n*(7+5*sqrt(5)))) / sqrt(5) - n. - Colin Barker, Mar 11 2017
a(n) = Fibonacci(n+3) + 3*Fibonacci(n+1) - n - 5. - G. C. Greubel, Jul 08 2019

cp's OEIS Frontend

A213580 Principal diagonal of the convolution array A213579.

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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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A053808 Partial sums of A001891.

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A213584 Rectangular array: (row n) = b**c, where b(h) = F(h+1), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

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A023552 Convolution of natural numbers >= 3 and Fibonacci numbers.

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GAP

Magma

Mathematica

PARI

PARI

Sage

Formula

A210730 a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=a(1)=0.

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GAP

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

A213584 Rectangular array: (row n) = bc, where b(h) = F(h+1), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.