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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A213753 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = -1 + 2^(n-1+h), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 3, 21, 16, 7, 58, 51, 36, 15, 141, 132, 111, 76, 31, 318, 307, 280, 231, 156, 63, 685, 672, 639, 576, 471, 316, 127, 1434, 1419, 1380, 1303, 1168, 951, 636, 255, 2949, 2932, 2887, 2796, 2631, 2352, 1911, 1276, 511, 5998, 5979, 5928, 5823
Offset: 1

Views

Author

Clark Kimberling, Jun 20 2012

Keywords

Comments

Principal diagonal: A213754.
Antidiagonal sums: A213755.
Row 1, (1,3,5,7,9,...)**(1,3,7,15,...): A047520.
Row 2, (1,3,5,7,9,...)**(3,7,15,31,...).
Row 3, (1,3,5,7,9,...)**(7,15,31,63...).
Ror a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....6.....21....58.....141
3....16....51....132....307
7....36....111...280....639
15...76....231...576....1303
31...156...471...1168...2631

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_] := 2 n - 1; 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}]  (* A213753 *)
Table[t[n, n], {n, 1, 40}] (* A213754 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213755 *)

Formula

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 + x + (-2 + 2^n)*x^2) and g(x) = (1 - 2*x)(1 - x )^3.

A368529 a(n) = Sum_{k=1..n} k^2 * 4^(n-k).

Original entry on oeis.org

0, 1, 8, 41, 180, 745, 3016, 12113, 48516, 194145, 776680, 3106841, 12427508, 49710201, 198841000, 795364225, 3181457156, 12725828913, 50903315976, 203613264265, 814453057460, 3257812230281, 13031248921608, 52124995686961, 208499982748420, 833999930994305
Offset: 0

Views

Author

Seiichi Manyama, Dec 28 2023

Keywords

Links

Crossrefs

Cf. A000290, A000330, A047520, A368528.
Cf. A052161, A368524.
Cf. A014825, A368530.

Programs

  • Mathematica
    LinearRecurrence[{7, -15, 13, -4}, {0, 1, 8, 41}, 30] (* Paolo Xausa, Jan 29 2024 *)
  • PARI
    a(n) = sum(k=1, n, k^2*4^(n-k));

Formula

G.f.: x * (1+x)/((1-4*x) * (1-x)^3).
a(n) = 7*a(n-1) - 15*a(n-2) + 13*a(n-3) - 4*a(n-4).
a(n) = A052161(n-1) + A052161(n-2) for n > 1.
a(n) = (5*4^(n+1) - (9*n^2 + 24*n + 20))/27.
a(0) = 0; a(n) = 4*a(n-1) + n^2.

A218376 a(n) = 5^n*sum_{i=1..n} i^5/5^i.

Original entry on oeis.org

0, 1, 37, 428, 3164, 18945, 102501, 529312, 2679328, 13455689, 67378445, 337053276, 1685515212, 8427947353, 42140274589, 210702132320, 1053511710176, 5267559970737, 26337801743253, 131689011192364, 658445059161820
Offset: 0

Views

Author

Robert G. Wilson v, Nov 28 2012

Keywords

Crossrefs

Cf. A000217, A047520, A066999, A067534. A008292.

Programs

  • Mathematica
    f[n_] := 5^n*Sum[i^5/5^i, {i, n}]; Array[f, 30, 0]

Formula

From Peter Bala, Nov 29 2012: (Start)
a(n) = 1/512*(3535*5^n - (128*n^5 + 800*n^4 + 2400*n^3 + 4600*n^2 + 5700*n + 3535)).
Recurrence equation: a(n) = 5*a(n-1) + n^5.
G.f.: (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)/((1 - 5*x)*(1 - x)^6) = x + 37*x^2 + 428*x^3 + ....
(End)

A368504 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * j^k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 11, 21, 10, 1, 0, 1, 20, 60, 58, 15, 1, 0, 1, 37, 161, 244, 141, 21, 1, 0, 1, 70, 428, 900, 857, 318, 28, 1, 0, 1, 135, 1149, 3164, 4225, 2787, 685, 36, 1, 0, 1, 264, 3132, 10990, 18945, 18196, 8704, 1434, 45, 1
Offset: 0

Views

Author

Seiichi Manyama, Dec 27 2023

Keywords

Examples

			Square array begins:
  1,  0,   0,    0,     0,      0,      0, ...
  1,  1,   1,    1,     1,      1,      1, ...
  1,  3,   6,   11,    20,     37,     70, ...
  1,  6,  21,   60,   161,    428,   1149, ...
  1, 10,  58,  244,   900,   3164,  10990, ...
  1, 15, 141,  857,  4225,  18945,  81565, ...
  1, 21, 318, 2787, 18196, 102501, 536046, ...

Links

Crossrefs

Columns k=0..5 give A000012, A000217, A047520, A066999, A067534, A218376.
Main diagonal gives A368505.
Cf. A368486.

Programs

  • PARI
    T(n, k) = sum(j=0, n, k^(n-j)*j^k);

Formula

G.f. of column k: x*A_k(x)/((1-k*x) * (1-x)^(k+1)), where A_n(x) are the Eulerian polynomials for k > 0.
T(0,k) = 0^k; T(n,k) = k*T(n-1,k) + n^k.

A368528 a(n) = Sum_{k=1..n} k^2 * 3^(n-k).

Original entry on oeis.org

0, 1, 7, 30, 106, 343, 1065, 3244, 9796, 29469, 88507, 265642, 797070, 2391379, 7174333, 21523224, 64569928, 193710073, 581130543, 1743391990, 5230176370, 15690529551, 47071589137, 141214767940, 423644304396, 1270932913813, 3812798742115
Offset: 0

Views

Author

Seiichi Manyama, Dec 28 2023

Keywords

Links

Crossrefs

Cf. A000290, A000330, A047520, A368529.
Cf. A052150, A368524.

Programs

  • PARI
    a(n) = sum(k=1, n, k^2*3^(n-k));

Formula

G.f.: x * (1+x)/((1-3*x) * (1-x)^3).
a(n) = 6*a(n-1) - 12*a(n-2) + 10*a(n-3) - 3*a(n-4).
a(n) = A052150(n-1) + A052150(n-2) for n > 1.
a(n) = (3^(n+1) - (n^2 + 3*n + 3))/2.
a(0) = 0; a(n) = 3*a(n-1) + n^2.
Previous Showing 11-15 of 15 results.

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