A213500 -id:A213500 - cp's OEIS frontend

A213570 Antidiagonal sums of the convolution array A213566.

1, 9, 37, 110, 272, 598, 1213, 2323, 4265, 7588, 13184, 22500, 37881, 63125, 104381, 171602, 280896, 458330, 746085, 1212415, 1967761, 3190824, 5170752, 8375400, 13561777, 21954753, 35536213, 57512918, 93073520, 150613438

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 (5,-9,6,1,-3,1).

Cf. A213566, A213500.

List([1..35], n-> Fibonacci(n+9)+Lucas(1,-1,n+8)[2] -(n^3+9*n^2 +39*n+81)); # G. C. Greubel, Jul 26 2019

[Fibonacci(n+9) +Lucas(n+8) -(n^3+9*n^2+39*n+81): n in [1..35]]; // 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 *)
Table[Fibonacci[n+9] + LucasL[n+8] -(n^3+9*n^2+39*n+81), {n,35}] (* G. C. Greubel, Jul 26 2019 *)

vector(35,n, f=fibonacci; 2*f(n+9)+f(n+7) -(n^3+9*n^2+39*n+81)) \\ G. C. Greubel, Jul 26 2019

[fibonacci(n+9) +lucas_number2(n+8,1,-1) -(n^3+9*n^2+39*n+81) for n in (1..35)] # G. C. Greubel, Jul 26 2019

a(n) = 5*a(n-1) - 9*a(n-2) + 6*a(n-3) + a(n-4) - 3*a(n-5) + a(n-6).
G.f.: f(x)/g(x), where f(x) = x*(1 + 4*x + x^2) and g(x) = (1 - x - x^2)*(1 - x)^4.
a(n) = Fibonacci(n+9) + Lucas(n+8) - n*(n^2 + 9*n + 39) - 81. - Ehren Metcalfe, Jul 10 2019
a(n) = Sum_{k=1..n} k^3 * Fibonacci(n+1-k). - Greg Dresden, Feb 27 2022

A213572 Principal diagonal of the convolution array A213571.

Original entry on oeis.org

1, 13, 82, 406, 1809, 7659, 31588, 128476, 518611, 2084809, 8361918, 33497010, 134094757, 536608663, 2146926472, 8588754808, 34357247847, 137433710421, 549744803650, 2199000186670, 8796044787481, 35184271425283

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 (11,-47,101,-116,68,-16).

Cf. A213571, A213500.

List([1..30], n-> 2^(2*n+1) -2^n*(n+2) -Binomial(n+1, 2)); # G. C. Greubel, Jul 25 2019

[2^(2*n+1) -2^n*(n+2) -Binomial(n+1, 2): n in [1..30]]; // 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^(2*n+1) -2^n*(n+2)-Binomial[n+1, 2], {n,30}] (* G. C. Greubel, Jul 25 2019 *)

vector(30, n, 2^(2*n+1) -2^n*(n+2) -binomial(n+1, 2)) \\ G. C. Greubel, Jul 25 2019

[2^(2*n+1) -2^n*(n+2) -binomial(n+1, 2) for n in (1..30)] # G. C. Greubel, Jul 25 2019

a(n) = (2^(n+2)*(2^n-1) - (2^(n+1) + n + 1)*n)/2.
a(n) = 11*a(n-1) - 47*a(n-2) + 101*a(n-3) - 116*a(n-4) + 68*a(n-5) - 16*a(n-6).
G.f.:  f(x)/g(x), where f(x) = x*(1 + 2*x - 14*x^2 + 14*x^3) and g(x) = (1 - 4*x)*((1 - x)^3)*(1 - 2*x)^2.

A213573 Rectangular array: (row n) = bc, where b(h) = 2^(h-1), c(h) = (n-1+h)^2, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 6, 4, 21, 17, 9, 58, 50, 34, 16, 141, 125, 93, 57, 25, 318, 286, 222, 150, 86, 36, 685, 621, 493, 349, 221, 121, 49, 1434, 1306, 1050, 762, 506, 306, 162, 64, 2949, 2693, 2181, 1605, 1093, 693, 405, 209, 81, 5998, 5486, 4462, 3310, 2286, 1486

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal: A213574.
Antidiagonal sums: A213575.
row 1, (1,2,4,8,...)**(1,4,9,16...): A047520.
row 2, (1,2,4,8,...)**(4,9,16,25...).
row 3, (1,2,4,8,...)**(9,16,25,36...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
   1,    6,   21,   58,  141,  318, ...
   4,   17,   50,  125,  286,  621, ...
   9,   34,   93,  222,  493, 1050, ...
  16,   57,  150,  349,  762, 1605, ...
  25,   86,  221,  506, 1093, 2286, ...
  36,  121,  306,  693, 1486, 3093, ...
  ...

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

Cf. A213500.

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

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

(* First program *)
b[n_]:= 2^(n-1); 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, 60}] (* A213573 *)
d = Table[T[n, n], {n, 40}] (* A213574 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213575 *)
(* Additional programs *)
Table[2^(n-k+1)*((k+1)^2 +2)-((n+2)^2 +2), {n,12}, {k, n}]//Flatten (* G. C. Greubel, Jul 25 2019 *)

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

[[2^(n-k+1)*((k+1)^2 +2)-((n+2)^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) = n^2 - (2*n^2 - 2*n - 1)*x + (n - 1)*x^2 and g(x) = (1 - 2*x)*(1 - x)^3.
T(n,k) = 2^k*(n^2 + 2*n + 3) - (n + k + 2)^2 + 2*(n + k + 1) - 1. - G. C. Greubel, Jul 25 2019

A213574 Principal diagonal of the convolution array A213573.

Original entry on oeis.org

1, 17, 93, 349, 1093, 3093, 8221, 20957, 51861, 125509, 298477, 699789, 1621285, 3718325, 8453181, 19069885, 42728245, 95156901, 210762253, 464517485, 1019214021, 2227173397, 4848613213, 10519312029, 22749902293, 49056576773, 105495131181, 226291086157

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 (9,-33,63,-66,36,-8).

Cf. A213568, A213500.

List([1..30], n-> 2^n*(3+2*n+n^2) - (3+4*n+4*n^2)); # G. C. Greubel, Jul 25 2019

[2^n*(3+2*n+n^2) - (3+4*n+4*n^2): n in [1..30]]; // G. C. Greubel, Jul 25 2019

(* First program *)
b[n_]:= 2^(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}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213568 *)
d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
(* Additional programs *)
LinearRecurrence[{9,-33,63,-66,36,-8},{1,17,93,349,1093,3093},30] (* Harvey P. Dale, Jun 25 2014 *)
Rest[CoefficientList[Series[x(1+8x-27x^2+10x^3+16x^4)/(1-3x+2x^2)^3, {x, 0, 30}], x]] (* Vincenzo Librandi, Jun 26 2014 *)

Vec(x*(1+8*x-27*x^2+10*x^3+16*x^4)/((1-x)^3*(1-2*x)^3) + O(x^30)) \\ Colin Barker, Oct 30 2017

vector(30, n, 2^n*(3+2*n+n^2) - (3+4*n+4*n^2)) \\ G. C. Greubel, Jul 25 2019

[2^n*(3+2*n+n^2) - (3+4*n+4*n^2) for n in (1..30)] # G. C. Greubel, Jul 25 2019

a(n) = 9*a(n-1) - 33*a(n-2) + 63*a(n-3) - 66*a(n-4) + 36*a(n-5) - 8*a(n-6).
G.f.: x*(1 + 8*x - 27*x^2 + 10*x^3 + 16*x^4)/(1 - 3*x + 2*x^2)^3.
a(n) = 2^n*(3+2*n+n^2) - (3+4*n+4*n^2). - Colin Barker, Oct 30 2017
E.g.f.: (3+6*x+4*x^2)*exp(2*x) - (3+8*x+4*x^2)*exp(x). - G. C. Greubel, Jul 25 2019

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

A213577 Principal diagonal of the convolution array A213576.

Original entry on oeis.org

1, 4, 17, 56, 172, 498, 1395, 3820, 10307, 27534, 73064, 193012, 508341, 1336132, 3507189, 9197732, 24107124, 63159782, 165433895, 433246860, 1134484871, 2970509594, 7777554192, 20363014056, 53312938537, 139578241348

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 (5,-6,-3,6,1,-1).

Cf. A213576, A213500.

List([1..40], n-> Fibonacci(2*n+3) - Fibonacci(n+3) - n*Fibonacci(n+1)); # G. C. Greubel, Jul 05 2019

[Fibonacci(2*n+3) -Fibonacci(n+3) -n*Fibonacci(n+1): n in [1..40]]; // G. C. Greubel, Jul 05 2019

(See A213576.)
LinearRecurrence[{5,-6,-3,6,1,-1},{1,4,17,56,172,498},30] (* Harvey P. Dale, Aug 23 2012 *)
Table[Fibonacci[2n+3] -Fibonacci[n+3] -n*Fibonacci[n+1], {n,1,40}] (* G. C. Greubel, Jul 05 2019 *)

vector(40, n, fibonacci(2*n+3) - fibonacci(n+3) - n*fibonacci(n+1)) \\ G. C. Greubel, Jul 05 2019

[fibonacci(2*n+3) - fibonacci(n+3) - n*fibonacci(n+1) for n in (1..40)] # G. C. Greubel, Jul 05 2019

a(n) = 5*a(n-1) - 6*a(n-2) - 3*a(n-3) + 6*a(n-4) + a(n-5) - a(n-6).
G.f.: x*(1 - x + 3*x^2 - 2*x^3)/((1 - 3*x + x^2)*(1 - x - x^2)^2).
a(n) = Fibonacci(2*n+3) - Fibonacci(n+3) - n*Fibonacci(n+1). - G. C. Greubel, Jul 05 2019

A213581 Antidiagonal sums of the convolution array A213571.

Original entry on oeis.org

1, 8, 36, 124, 367, 988, 2498, 6048, 14197, 32576, 73472, 163508, 360027, 785908, 1703294, 3669240, 7863393, 16776120, 35650300, 75495980, 159381831, 335542348, 704640826, 1476392464, 3087004877, 6442447728, 13421769208

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 (8,-26,44,-41,20,-4).

Cf. A213571, A213500.

List([1..35], n-> 2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6); # G. C. Greubel, Jul 26 2019

[2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6: n in [1..35]]; // G. C. Greubel, Jul 26 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 *)
(* Second program *)
Table[2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6, {n,35}] (* G. C. Greubel, Jul 26 2019 *)

vector(35, n, 2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6) \\ G. C. Greubel, Jul 26 2019

[2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6 for n in (1..35)] # G. C. Greubel, Jul 26 2019

a(n) = 8*a(n-1) - 26*a(n-2) + 44*a(n-3) - 41*a(n-4) + 20*a(n-5) - 4*a(n-6).
G.f.: f(x)/g(x), where f(x) = x*(1 - 2*x^2) and g(x) = (1 - x)^4*(1 - 2*x)^2.
a(n) = 8 +(n-2)*2^(n+2) -(n-2)*n*(n+5)/6. - Bruno Berselli, Jul 09 2012

A213582 Rectangular array: (row n) = bc, where b(h) = -1 + 2^h, c(h) = n-1+h, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 5, 2, 16, 9, 3, 42, 27, 13, 4, 99, 68, 38, 17, 5, 219, 156, 94, 49, 21, 6, 466, 339, 213, 120, 60, 25, 7, 968, 713, 459, 270, 146, 71, 29, 8, 1981, 1470, 960, 579, 327, 172, 82, 33, 9, 4017, 2994, 1972, 1207, 699, 384, 198, 93, 37, 10, 8100, 6053, 4007, 2474, 1454, 819, 441, 224, 104, 41, 11

Offset: 1

Clark Kimberling, Jun 19 2012

Principal diagonal: A213583.
Antidiagonal sums: A156928.
Row 1, (1,3,7,15,31,...)**(1,2,3,4,5,...): A002662.
Row 2, (1,3,7,15,31,...)**(2,3,4,5,6,...)
Row 3, (1,3,7,15,31,...)**(3,4,5,6,7,...)
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...5....16...42....99....219
2...9....27...68....156...339
3...13...38...94....213...459
4...17...49...120...270...579
5...21...60...146...327...699
6...25...71...172...384...819

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

Cf. A213500, A213571.

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

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

(* First program *)
b[n_]:= 2^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}]] (* A213582 *)
r[n_]:= Table[T[n, k], {k, 40}]
Table[T[n, n], {n, 1, 40}] (* A213583 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A156928 *)
(* Second program *)
Table[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)

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

[[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 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) = n - (n-1)*x and g(x) = (1-2*x) *(1-x)^3.
T(n,k) = 2*(n+1)*(2^k - 1) - k*(k + 2*n + 3)/2. - G. C. Greubel, Jul 08 2019

A213588 Principal diagonal of the convolution array A213587.

Original entry on oeis.org

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

cp's OEIS Frontend

A213570 Antidiagonal sums of the convolution array A213566.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A213572 Principal diagonal of the convolution array A213571.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A213573 Rectangular array: (row n) = b**c, where b(h) = 2^(h-1), c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A213574 Principal diagonal of the convolution array A213573.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

PARI

Sage

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A213577 Principal diagonal of the convolution array A213576.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

A213573 Rectangular array: (row n) = bc, where b(h) = 2^(h-1), c(h) = (n-1+h)^2, 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.

A213582 Rectangular array: (row n) = bc, where b(h) = -1 + 2^h, c(h) = n-1+h, n>=1, h>=1, and = convolution.