A053808 -id:A053808 - 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.

A192951 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 3, 9, 20, 40, 74, 131, 225, 379, 630, 1038, 1700, 2773, 4511, 7325, 11880, 19252, 31182, 50487, 81725, 132271, 214058, 346394, 560520, 906985, 1467579, 2374641, 3842300, 6217024, 10059410, 16276523, 26336025, 42612643, 68948766

Offset: 0

Clark Kimberling, Jul 13 2011

The titular polynomials are defined recursively:  p(n,x) = x*p(n-1,x) + 3n - 1, with p(0,x)=1.  For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.
...
The list of examples at A192744 is extended here; the recurrence is given by p(n,x) = x*p(n-1,x) + v(n), with p(0,x)=1, and the reduction of p(n,x) by x^2 -> x+1 is represented by u1 + u2*x:
...
If v(n)=        n, then u1=A001595, u2=A104161.
If v(n)=      n-1, then u1=A001610, u2=A066982.
If v(n)=     3n-1, then u1=A171516, u2=A192951.
If v(n)=     3n-2, then u1=A192746, u2=A192952.
If v(n)=     2n-1, then u1=A111314, u2=A192953.
If v(n)=      n^2, then u1=A192954, u2=A192955.
If v(n)=   -1+n^2, then u1=A192956, u2=A192957.
If v(n)=    1+n^2, then u1=A192953, u2=A192389.
If v(n)=   -2+n^2, then u1=A192958, u2=A192959.
If v(n)=    2+n^2, then u1=A192960, u2=A192961.
If v(n)=    n+n^2, then u1=A192962, u2=A192963.
If v(n)=   -n+n^2, then u1=A192964, u2=A192965.
If v(n)= n(n+1)/2, then u1=A030119, u2=A192966.
If v(n)= n(n-1)/2, then u1=A192967, u2=A192968.
If v(n)= n(n+3)/2, then u1=A192969, u2=A192970.
If v(n)=     2n^2, then u1=A192971, u2=A192972.
If v(n)=   1+2n^2, then u1=A192973, u2=A192974.
If v(n)=  -1+2n^2, then u1=A192975, u2=A192976.
If v(n)=  1+n+n^2, then u1=A027181, u2=A192978.
If v(n)=  1-n+n^2, then u1=A192979, u2=A192980.
If v(n)=  (n+1)^2, then u1=A001891, u2=A053808.
If v(n)=  (n-1)^2, then u1=A192981, u2=A192982.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (3, -2, -1, 1).

Cf. A000045, A192232, A192744.

F:=Fibonacci;; List([0..40], n-> F(n+4)+2*F(n+2)-(3*n+5)); # G. C. Greubel, Jul 12 2019

I:=[0, 1, 3, 9]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-1*Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Nov 16 2011

F:=Fibonacci; [F(n+4)+2*F(n+2)-(3*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + 3n - 1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A171516 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192951 *)
(* Additional programs *)
LinearRecurrence[{3,-2,-1,1},{0,1,3,9},40] (* Vincenzo Librandi, Nov 16 2011 *)
With[{F=Fibonacci}, Table[F[n+4]+2*F[n+2]-(3*n+5), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,-1,-2,3]^n*[0;1;3;9])[1,1] \\ Charles R Greathouse IV, Mar 22 2016

vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2)-(3*n+5)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [f(n+4)+2*f(n+2)-(3*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From Bruno Berselli, Nov 16 2011: (Start)
G.f.: x*(1+2*x^2)/((1-x)^2*(1 - x - x^2)).
a(n) = ((25+13*t)*(1+t)^n + (25-13*t)*(1-t)^n)/(10*2^n) - 3*n - 5 = A000285(n+2) - 3*n - 5  where t=sqrt(5). (End)
a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (3*n+5). - G. C. Greubel, Jul 12 2019

A011794 Triangle defined by T(n+1, k) = T(n, k-1) + T(n-1, k), T(n,1) = 1, T(1,k) = 1, T(2,k) = min(2,k).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 6, 7, 8, 1, 4, 7, 11, 12, 13, 1, 4, 10, 14, 19, 20, 21, 1, 5, 11, 21, 26, 32, 33, 34, 1, 5, 15, 25, 40, 46, 53, 54, 55, 1, 6, 16, 36, 51, 72, 79, 87, 88, 89, 1, 6, 21, 41, 76, 97, 125, 133, 142, 143, 144, 1, 7, 22, 57, 92, 148, 176, 212, 221, 231, 232, 233

Offset: 1

N. J. A. Sloane and David Broadhurst

			matrix(10,10,n,k,a(n-1,k-1))
  [ 0 0 0 0 0 0 0 0 0 0 ]
  [ 0 1 1 1 1 1 1 1 1 1 ]
  [ 0 1 2 2 2 2 2 2 2 2 ]
  [ 0 1 2 3 3 3 3 3 3 3 ]
  [ 0 1 3 4 5 5 5 5 5 5 ]
  [ 0 1 3 6 7 8 8 8 8 8 ]
Triangle begins as:
  1;
  1, 2;
  1, 2,  3;
  1, 3,  4,  5;
  1, 3,  6,  7,  8;
  1, 4,  7, 11, 12, 13;
  1, 4, 10, 14, 19, 20, 21;
  1, 5, 11, 21, 26, 32, 33, 34;
  1, 5, 15, 25, 40, 46, 53, 54, 55;
  1, 6, 16, 36, 51, 72, 79, 87, 88, 89;

G. C. Greubel, Rows n = 1..100 of the triangle, flattened
D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.

Columns include A008619 and (essentially) A055802, A055803, A055804, A055805, A055806.
Right-hand columns 1-14 are A000045, A000071, A001911, A001924, A001891, A014162, A053808, A014166, A053809, A053739, A054469, A053295, A054470, A053296.
Essentially a reflected version of A055801.
Sums include: A039834 (signed row), A131913 (row).
Cf. A008466, A074878, A103609.

function T(n,k) // T = A011794(n,k)
  if k eq 1 or n eq 1 then return 1;
  elif n eq 2 then return Min(2, k);
  else return T(n-1,k-1) + T(n-2,k);
  end if;
end function;
[T(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Oct 21 2024

T[n_, k_]:= T[n, k]= T[n-1, k-1] + T[n-2, k]; T[n_, 1] = 1; T[1, k_] = 1; T[2, k_] := Min[2, k]; Table[T[n, k], {n,15}, {k,n}]//Flatten (* Jean-François Alcover, Feb 26 2013 *)

T(n,k)=if(n<=0 || k<=0,0, if(n<=2 || k==1, min(n,k), T(n-1,k-1)+T(n-2,k)))

def T(n, k): # T = A011794
    if (k==1 or n==1): return 1
    elif (n==2): return min(2,k)
    else: return T(n-1, k-1) + T(n-2, k)
flatten([[T(n, k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Oct 21 2024

T(n,n) = Fibonacci(n+1). - Jean-François Alcover, Feb 26 2013
From G. C. Greubel, Oct 21 2024: (Start)
Sum_{k=1..n} T(n, k) = A131913(n-1).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A039834(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1,k) = (1/2)*((1-(-1)^n)*A074878((n+3)/2) + (1+(-1)^n)*A008466((n+6)/2)) (diagonal row sums).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1,k) = (-1)^floor((n-1)/2)*A103609(n) + [n=1] (signed diagonal row sums). (End)

Entry improved by comments from Michael Somos

More terms added by G. C. Greubel, Oct 21 2024

A213579 Rectangular array: (row n) = bc, where b(h) = F(h), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 3, 2, 7, 5, 3, 14, 11, 7, 4, 26, 21, 15, 9, 5, 46, 38, 28, 19, 11, 6, 79, 66, 50, 35, 23, 13, 7, 133, 112, 86, 62, 42, 27, 15, 8, 221, 187, 145, 106, 74, 49, 31, 17, 9, 364, 309, 241, 178, 126, 86, 56, 35, 19, 10, 596, 507, 397, 295, 211, 146, 98, 63, 39, 21

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal: A213580.
Antidiagonal sums: A053808.
Row 1, (1,1,2,3,5,...)**(1,2,3,4,...): A001924.
Row 2, (1,1,2,3,5,...)**(2,3,4,5,...): A023548.
Row 3, (1,1,2,3,5,...)**(3,4,5,6,...): A023552.
Row 4, (1,1,2,3,5,...)**(4,5,6,7,...): A210730.
Row 5, (1,1,2,3,5,...)**(5,6,7,8,...): A210731.
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....3....7....14...26...46
2....5....11...21...38...66
3....7....15...28...50...86
4....9....19...35...62...106
5....11...23...42...74...126
6....13...27...49...86...146

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

Cf. A000045, A213500.

Flat(List([1..12], n-> List([1..n], k-> Fibonacci(k+3) + n*Fibonacci(k+2) -(n+k+2) ))); # G. C. Greubel, Jul 08 2019

[[Fibonacci(k+3) + n*Fibonacci(k+2) -(n+k+2): k in [1..n]]: n in [1..12]]; // 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-k+4] +k*Fibonacci[n-k+3] -(n+3), {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)

t(n,k) = fibonacci(n-k+4) + k*fibonacci(n-k+3) - (n+3);
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019

[[fibonacci(k+3) + n*fibonacci(k+2) -(n+k+2) 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 - (n-1)*x and g(x) = (1-x-x^2) *(1-x)^2.
T(n, k) = Fibonacci(k+3) + n*Fibonacci(k+2) - (n+k+2). - G. C. Greubel, Jul 08 2019

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

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

A213580 Principal diagonal of the convolution array A213579.

Original entry on oeis.org

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

A053809 Second partial sums of A001891.

Original entry on oeis.org

1, 6, 21, 57, 133, 281, 554, 1039, 1878, 3302, 5686, 9638, 16143, 26796, 44179, 72471, 118435, 193015, 313920, 509805, 827036, 1340636, 2171996, 3517532, 5695053, 9218786, 14920769, 24147269, 39076593, 63233317, 102320326

Offset: 0

Barry E. Williams, Mar 27 2000

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
Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

Cf. A001911, A001891, A053808.

Right-hand column 9 of triangle A011794. Pairwise sums of A014166.

List([0..40], n-> Fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6) # G. C. Greubel, Jul 06 2019

[Fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6: n in [0..40]]; // G. C. Greubel, Jul 06 2019

Table[Fibonacci[n+10] - (2*n^3+27*n^2+145*n+324)/6, {n,0,40}] (* G. C. Greubel, Jul 06 2019 *)

vector(40, n, n--; fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6) \\ G. C. Greubel, Jul 06 2019

[fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6 for n in (0..40)] # G. C. Greubel, Jul 06 2019

a(n) = a(n-1) + a(n-2) + (2*n+3)*C(n+2, 2)/3; a(-x)=0.
a(n) = Fibonacci(n+10) - (2*n^3 + 27*n^2 + 145*n + 324)/6.
G.f.: (1+x)/((1-x)^4*(1-x-x^2)).
a(n) = 5*a(n-1) - 9*a(n-2) + 6*a(n-3) + a(n-4) - 3*a(n-5) + a(n-6). - Wesley Ivan Hurt, Apr 21 2021

A163250 a(n) = A000045(n+6) - (n^2 + 4*n + 8).

Original entry on oeis.org

0, 0, 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

Offset: 0

Jonathan Vos Post, Jul 23 2009

Given on p. 2 of Freixas, and proved as Theorem 3.2.
Partial sums of A001891. - Bill McEachen, Jan 20 2023
Original name was: The number of nonisomorphic complete simple games with n voters of two different types. - Charles R Greathouse IV, Jan 22 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
Josep Freixas, Xavier Molinero, and Salvador Roura, A Fibonacci sequence for linear structures with two types of components, arXiv:0907.3853 [math.CO], Jul 22 2009.
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A000045, A053808, A001891.

List([0..35],n->Fibonacci(n+6)-(n^2+4*n+8)); # Muniru A Asiru, Oct 28 2018

[Fibonacci(n+6)-(n^2+4*n+8): n in [0..40]]; // Vincenzo Librandi, Sep 22 2017

with(numtheory): seq(coeff(series(x^2*(1+x)/((x^2+x-1)*(x-1)^3),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 28 2018

LinearRecurrence[{4,-5,1,2,-1}, {0,0,1,5,15}, 40] (* or *) Table[ Fibonacci[n+6] -(n^2+4*n+8), {n,0,40}] (* G. C. Greubel, Dec 12 2016 *)

concat([0,0], Vec(x^2*(1+x)/((1-x-x^2)*(1-x)^3) + O(x^40))) \\ G. C. Greubel, Dec 12 2016

f=fibonacci; [f(n+6) -(n^2+4*n+8) for n in (0..40)] # G. C. Greubel, Jul 06 2019

a(n) = F(n+6) - (n^2 + 4*n + 8), where F(n) are the Fibonacci numbers.
From R. J. Mathar, Jul 27 2009: (Start)
a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5).
G.f.: x^2*(1+x)/((1-x-x^2)*(1-x)^3). (End)
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} i^2 * C(n-k-1,k-i). - Wesley Ivan Hurt, Sep 21 2017
a(n) = A053808(n-2) for n >= 2. - Georg Fischer, Oct 28 2018
a(n) = (n-1)^2 + a(n-1) + a(n-2), n>2 (conjectured). - Bill McEachen, Jan 20 2023

More terms from R. J. Mathar, Jul 27 2009

New name using given formula from Joerg Arndt, Jan 21 2023

A220887 Number CG(n,3) of complete games with n players belonging to 3 types.

Original entry on oeis.org

6, 50, 262, 1114, 4278, 15769, 58147, 221089, 886411, 3806475, 17681979, 89337562, 492188528, 2959459154, 19424078142, 139141985438, 1087614361775, 9274721292503

Offset: 4

N. J. A. Sloane, Dec 29 2012

J. Freixas and S. Kurz, The golden number and Fibonacci sequences in the design of voting structures, 2012.
J. Freixas and S. Kurz, The golden number and Fibonacci sequences in the design of voting structures, arXiv:1401.8180 [math.CO], 2014.

Cf. A053808.

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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A192951 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A011794 Triangle defined by T(n+1, k) = T(n, k-1) + T(n-1, k), T(n,1) = 1, T(1,k) = 1, T(2,k) = min(2,k).

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A213579 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), 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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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.

A213579 Rectangular array: (row n) = bc, where b(h) = F(h), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), 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.

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.