cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A212891 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 4, 20, 17, 9, 50, 46, 34, 16, 105, 100, 84, 57, 25, 196, 190, 170, 134, 86, 36, 336, 329, 305, 260, 196, 121, 49, 540, 532, 504, 450, 370, 270, 162, 64, 825, 816, 784, 721, 625, 500, 356, 209, 81, 1210, 1200, 1164, 1092, 980, 830, 650, 454, 262
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2012

Keywords

Comments

Principal diagonal: A213436
Antidiagonal sums: A024166
row 1, (1,2,3,...)**(1,4,9,...): A002415(k+1)
row 2, (1,2,3,...)**(4,9,16,...): k*(k^3 + 8*k^2 + 23*k + 16)/12
row 3, (1,2,3,...)**(9,16,25,...): k*(k^3 + 12*k^2 + 53*k + 42)/12
...
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....6....20....50....105....196...336
4....17...46....100...190....329...532
9....34...84....170...305....504...784
16...57...134...260...450....721...1092
25...86...196...370...625....980...1456
...
T(5,1) = (1)**(25) = 25
T(5,2) = (1,2)**(25,36) = 1*36+2*25 = 86
T(5,3) = (1,2,3)**(25,36,49) = 1*49+2*36+3*25 = 196

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_] := 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}]  (* A212891 *)
d = Table[t[n, n], {n, 1, 40}] (* A213436 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A024166  *)

Formula

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

A213504 Principal diagonal of the convolution array A213590.

Original entry on oeis.org

1, 6, 35, 138, 488, 1564, 4733, 13734, 38711, 106846, 290496, 781264, 2084753, 5531846, 14619811, 38527834, 101328712, 266119228, 698218525, 1830665830, 4797572551, 12568780126, 32920653120, 86214096768, 225758326273
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2012

Keywords

Links

Crossrefs

Cf. A213590, A213500.

Programs

  • GAP
    F:=Fibonacci;; List([1..40], n-> F(2*n+6) -F(n+6) -2*n*F(n+3) -n^2*F(n+1)); # G. C. Greubel, Jul 06 2019
  • Magma
    F:=Fibonacci; [F(2*n+6) -F(n+6) -2*n*F(n+3) -n^2*F(n+1): n in [1..40]]; // G. C. Greubel, Jul 06 2019
  • Mathematica
    (* 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 *)
With[{F = Fibonacci}, Table[F[2*n+6] -F[n+6] -2*n*F[n+3] -n^2*F[n+1], {n, 40}]] (* G. C. Greubel, Jul 06 2019 *)
  • PARI
    vector(40, n, my(f=fibonacci); f(2*n+6) - f(n+6) - 2*n*f(n+3) - n^2*f(n+1)) \\ G. C. Greubel, Jul 06 2019
  • Sage
    f=fibonacci; [f(2*n+6) -f(n+6) -2*n*f(n+3) -n^2*f(n+1) for n in (1..40)] # G. C. Greubel, Jul 06 2019

Formula

a(n) = 6*a(n-1) - 10*a(n-2) - 2*a(n-3) + 15*a(n-4) - 2*a(n-5)- 8*a(n-6) + a(n-8).
G.f.: x*(1 + 9*x^2 - 10*x^3 + 7*x^4 - 2*x^5)/((1 - 3*x + x^2)*(1 - x - x^2)^3). [corrected by Georg Fischer, May 11 2019]
a(n) = Fibonacci(2*n+6) - Fibonacci(n+6) - 2*n*Fibonacci(n+3) - n^2*Fibonacci(n+1). - G. C. Greubel, Jul 06 2019

A213505 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 8, 4, 34, 25, 9, 104, 88, 52, 16, 259, 234, 170, 89, 25, 560, 524, 424, 280, 136, 36, 1092, 1043, 899, 674, 418, 193, 49, 1968, 1904, 1708, 1384, 984, 584, 260, 64, 3333, 3252, 2996, 2555, 1979, 1354, 778, 337, 81, 5368, 5268, 4944, 4368, 3584
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2012

Keywords

Comments

Principal diagonal: A213546.
Antidiagonal sums: A213547.
Row 1, (1,4,9,...)**(1,4,9,...): A033455.
Row 2, (1,4,9,...)**(4,9,16,...): (k^5 + 10*k^4 + 40*k^3 + 50*k^2 +19*k)/30.
Row 3, (1,4,9,...)**(9,16,25,...): (k^5 + 15*k^4 + 90*k^3 + 120*k^2+44*k)/30.
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....8.....34....104...259....560
4....25....88....234...524....1043
9....52....170...424...899....1708
16...89....280...674...1384...2555
25...136...418...984...1979...3584
...
T(5,1) = (1)**(25) = 25
T(5,2) = (1,4)**(25,36) = 1*36+4*25 = 136
T(5,3) = (1,4,9)**(25,36,49) = 1*49+4*36+9*25 = 418

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_] := n^2; 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}]  (* A213505 *)
d = Table[t[n, n], {n, 1, 40}] (* A213546 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213547 *)

Formula

T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) - T(n,k-6).
G.f. for row n: f(x)/g(x), where f(x) = n^2 - (n^2 - 2*n - 1)*x - (n^2 - 2)*x^2 - ((n - 1)^2)*x^3 and g(x) = (1 - x)^6.

A213551 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 3, 21, 15, 6, 56, 46, 28, 10, 126, 111, 81, 45, 15, 252, 231, 186, 126, 66, 21, 462, 434, 371, 281, 181, 91, 28, 792, 756, 672, 546, 396, 246, 120, 36, 1287, 1242, 1134, 966, 756, 531, 321, 153, 45, 2002, 1947, 1812, 1596, 1316, 1001, 686, 406
Offset: 1

Views

Author

Clark Kimberling, Jun 17 2012

Keywords

Comments

Principal diagonal: A213552
Antidiagonal sums: A051923
Row 1, (1,3,6,...)**(1,3,6,...): A000389
Row 2, (1,3,6,...)**(3,6,10,...): (k^5 + 15*k^4 + 85*k^3 + 165*k^2 + 94*k)/120
Row 3, (1,3,6,...)**(6,10,15,...): (k^5 + 20*k^4 + 155*k^3 + 340*k^2 + 204*k)/120
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....6....21....56....126....252
3....15...46....111...231....434
6....28...81....186...371....672
10...45...126...281...546....966
15...66...181...396...756....1316
21...91...246...531...1001...1722

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_] := n (n + 1)/2; c[n_] := n (n + 1)/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}]  (* A213551 *)
d = Table[t[n, n], {n, 1, 40}] (* A213552 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A051923 *)

Formula

T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) - T(n,k-6).
G.f. for row n: f(x)/g(x), where f(x) = n*(n+1) - 2*((n-1)^2)*x + 2*(n-1)*x^2 and g(x) = 2*(1 - x)^2.

A213555 Rectangular array: (row n) = b**c, where b(h) = h^3, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 10, 2, 46, 19, 3, 146, 82, 28, 4, 371, 246, 118, 37, 5, 812, 596, 346, 154, 46, 6, 1596, 1253, 821, 446, 190, 55, 7, 2892, 2380, 1694, 1046, 546, 226, 64, 8, 4917, 4188, 3164, 2135, 1271, 646, 262, 73, 9, 7942, 6942, 5484, 3948, 2576, 1496, 746
Offset: 1

Views

Author

Clark Kimberling, Jun 17 2012

Keywords

Comments

Principal diagonal: A213556.
Antidiagonal sums: A213547.
Row 1, (1,8,27,...)**(1,2,3,...): A024166.
Row 2, (1,8,27,...)**(2,3,4,...): (3*k^5 + 30*k^4 + 55*k^3 + 30*k^2 + 2*k)/60.
Row 3, (1,8,27,...)**(3,4,5,...): (3*k^5 + 45*k^4 + 85*k^3 + 45*k^2 + 2*k)/60.
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1...10...46....146...371....812
2...19...82....246...596....1253
3...28...118...346...821....1694
4...37...154...446...1046...2135
5...46...190...546...1271...2576
6...55...226...646...1496...3017

Links

Crossrefs

Cf. A213500, A213553.

Programs

  • Mathematica
    b[n_] := n^3; 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}]  (* A213555 *)
d = Table[t[n, n], {n, 1, 40}] (* A213556 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213547 *)

Formula

T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) -T(n,k-6).
G.f. for row n: f(x)/g(x), where f(x) = n + (3*n + 1)*x - (3*n - 4)*x^2 - (n - 1)*x^3 and g(x) = (1 - x)^6.

A213557 Antidiagonal sums of the convolution array A213590.

Original entry on oeis.org

1, 6, 23, 70, 184, 438, 971, 2042, 4125, 8076, 15424, 28876, 53189, 96670, 173747, 309362, 546456, 958690, 1672015, 2901170, 5011321, 8621976, 14781888, 25263000, 43053769, 73186038, 124119311, 210055582, 354806200, 598245006
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2012

Keywords

Links

Crossrefs

Cf. A213590, A213500.

Programs

  • GAP
    F:=Fibonacci;; List([1..40], n-> n*F(n+7) -2*F(n+9) +2*(n^2+10*n+ 34)); # G. C. Greubel, Jul 06 2019
  • Magma
    F:=Fibonacci; [n*F(n+7) -2*F(n+9) +2*(n^2+10*n+34): n in [1..40]]; // G. C. Greubel, Jul 06 2019
  • Mathematica
    (* 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 *)
With[{F = Fibonacci}, Table[n*F[n+7] -2*F[n+9] +2*(n^2+10*n+34), {n,40}]] (* G. C. Greubel, Jul 06 2019 *)
LinearRecurrence[{5,-8,2,6,-4,-1,1},{1,6,23,70,184,438,971},30] (* Harvey P. Dale, Jun 04 2025 *)
  • PARI
    vector(40, n, f=fibonacci; n*f(n+7) -2*f(n+9) +2*(n^2+10*n+34)) \\ G. C. Greubel, Jul 06 2019
  • Sage
    f=fibonacci; [n*f(n+7) -2*f(n+9) +2*(n^2+10*n+34) for n in (1..40)] # G. C. Greubel, Jul 06 2019

Formula

a(n) = 5*a(n-1) - 8*a(n-2) + 2*a(n-3) + 6*a(n-4) - 4*a(n-5) - a(n-6) + a(n-7).
G.f.: f(x)/g(x), where f(x) = x*(1 + x + x^2 + x^3) and g(x) = (1 - x)^3 (1 - x - x^2)^2.
a(n) = n*Fibonacci(n+7) - 2*Fibonacci(n+9) + 2*n^2 + 20*n + 68. - G. C. Greubel, Jul 06 2019

A213558 Rectangular array: (row n) = b**c, where b(h) = h^3, c(h) = (n-1+h)^3, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 16, 8, 118, 91, 27, 560, 496, 280, 64, 2003, 1878, 1366, 637, 125, 5888, 5672, 4672, 2944, 1216, 216, 14988, 14645, 12917, 9542, 5446, 2071, 343, 34176, 33664, 30920, 25088, 17088, 9088, 3256, 512, 71445, 70716, 66620, 57359, 43535
Offset: 1

Views

Author

Clark Kimberling, Jun 17 2012

Keywords

Comments

Principal diagonal: A213559
Antidiagonal sums: A213560
Row 1, (1,8,27,...)**(1,8,27,...): A145216
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1.....16.....118....560.....2003
8.....91.....496....1878....5672
27....280....1366...4672....12917
64....637....2944...9542....25088
125...1216...5446...17088...43535

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_] := n^3; c[n_] := n^3
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}]  (* A213558 *)
d = Table[t[n, n], {n, 1, 40}] (* A213559 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213560 *)

Formula

T(n,k) = 8*T(n,k-1) - 28*T(n,k-2) + 56*T(n,k-3) - 70*T(n,k-4) + 56*T(n,k-5) - 28*T(n,k-6) + 8*T(n,k-7) - T(n,k-8).
G.f. for row n: f(x)/g(x), where f(x) = n^3 + ((n + 1)^3)*x + (-8*n^3 + 6*n^2 + 12*n + 8)*x^2 + (8*n^3 - 18*n^2 + 18)*x^3 - ((n - 2)^3)*x^4 - ((n + 1)^3)*x^5 and g(x) = (1 - x)^8.

A213561 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = m*(m+1)/2, m=n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 7, 3, 27, 18, 6, 77, 61, 34, 10, 182, 157, 109, 55, 15, 378, 342, 267, 171, 81, 21, 714, 665, 557, 407, 247, 112, 28, 1254, 1190, 1043, 827, 577, 337, 148, 36, 2079, 1998, 1806, 1512, 1152, 777, 441, 189, 45, 3289, 3189, 2946, 2562, 2072, 1532
Offset: 1

Views

Author

Clark Kimberling, Jun 18 2012

Keywords

Comments

Principal diagonal: A213562
Antidiagonal sums: A213563
Row 1, (1,4,9,...)**(1,3,6,...): A005585
Row 2, (1,4,9,...)**(3,6,10,...): (2*k^5 +25*k^4 + 120*k^3 + 155*k^2 + 58*k)/120
Row 3, (1,4,9,...)**(6,10,15,...): (2*k^5 +35*k^4 + 60*k^3 + 325*k^2 + 118*k)/120
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....7.....27....77....182
3....18....61....157...342
6....34....109...267...557
10...55....171...407...827
15...81....247...577...1152
21...112...337...777...1532

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_] := n^2; c[n_] := n (n + 1)/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}]  (* A213561 *)
d = Table[t[n, n], {n, 1, 40}] (* A213562 *)
s1 = Table[s[n], {n, 1, 50}] (* A213563 *)

Formula

T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) - T(n,k-6).
G.f. for row n: f(x)/g(x), where f(x) = n*(n + 1) - (n^2 - n - 2)*x - (n^2 + n - 2)*x^2 + n*(n - 1)*x^3 and g(x) = 2*(1 - x)^6.

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.

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

Views

Author

Clark Kimberling, Jun 19 2012

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Cf. A213500, A213587.

Programs

  • GAP
    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
  • Magma
    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
  • Mathematica
    (* 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 *)
  • PARI
    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
  • Sage
    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

Formula

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

A213567 Principal diagonal of the convolution array A213566.

Original entry on oeis.org

1, 13, 59, 183, 476, 1108, 2409, 4993, 10007, 19559, 37504, 70832, 132145, 244029, 446763, 811847, 1465676, 2630836, 4697945, 8350305, 14779671, 26058903, 45784224, 80179968, 139995361, 243755533, 423324539, 733409943
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2012

Keywords

Links

Crossrefs

Cf. A213566, A213500.

Programs

  • GAP
    F:=Fibonacci;; List([1..30], n-> (2*n+3)*F(n+3)+(n^2+2)*F(n+2) -4*(n^2+2*n+2)); # G. C. Greubel, Jul 26 2019
  • Magma
    F:= Fibonacci; [(2*n+3)*F(n+3)+(n^2+2)*F(n+2) -4*(n^2+2*n+2): n in [1..30]]; // G. C. Greubel, Jul 26 2019
  • Mathematica
    (* 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[(2*n+3)*Fibonacci[n+3] +(n^2+2)*Fibonacci[n+2] -4*(n^2+2*n+2), {n, 30}] (* G. C. Greubel, Jul 26 2019 *)
  • PARI
    vector(30, n, f=fibonacci; (2*n+3)*f(n+3)+(n^2+2)*f(n+2) -4*(n^2+ 2*n+2)) \\ G. C. Greubel, Jul 26 2019
  • Sage
    f=fibonacci; [(2*n+3)*f(n+3)+(n^2+2)*f(n+2) -4*(n^2+ 2*n+2) for n in (1..30)] # G. C. Greubel, Jul 26 2019

Formula

a(n) = 6*a(n-1) - 12*a(n-2) - 5*a(n-3) + 12*a(n-4) - 12*a(n-5) - 3*a(n-6) + 6*a(n-7) - a(n-9).
G.f.: f(x)/g(x), where f(x) = x*(1 + 7*x - 7*x^2 - 20*x^3 + 9*x^4 + 9*x^5 + 9*x^6) and g(x) = (1 - 2*x + x^3)^3.
a(n) = (2*n + 3)*Fibonacci(n+3) + (n^2 + 2)*Fibonacci(n+2) - 4*(n^2 + 2*n + 2). - G. C. Greubel, Jul 26 2019
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