A094195 -id:A094195 - cp's OEIS frontend

A094250 Array, A(n, k) = ((n+2)^(k+1) + (k+1)n(n+1) - 1)/(n+1)^2, read by antidiagonals.

1, 1, 3, 1, 3, 7, 1, 3, 8, 15, 1, 3, 9, 22, 31, 1, 3, 10, 31, 63, 63, 1, 3, 11, 42, 117, 185, 127, 1, 3, 12, 55, 199, 459, 550, 255, 1, 3, 13, 70, 315, 981, 1825, 1644, 511, 1, 3, 14, 87, 471, 1871, 4888, 7287, 4925, 1023, 1, 3, 15, 106, 673, 3273, 11203, 24420, 29133, 14767, 2047

Offset: 0

N. J. A. Sloane, Jun 02 2004

			Array, A(n, k), begins:
  1, 3,  7, 15,  31,   63,   127,    255,     511, ... A000225;
  1, 3,  8, 22,  63,  185,   550,   1644,    4925, ... A047926;
  1, 3,  9, 31, 117,  459,  1825,   7287,   29133, ... A073724;
  1, 3, 10, 42, 199,  981,  4888,  24420,  122077, ... A094195;
  1, 3, 11, 55, 315, 1871, 11203,  67191,  403115, ... A094259;
  1, 3, 12, 70, 471, 3273, 22882, 160140, 1120941, ...
Antidiagonals, T(n, k), begins as:
  1;
  1, 3;
  1, 3,  7;
  1, 3,  8, 15;
  1, 3,  9, 22,  31;
  1, 3, 10, 31,  63,   63;
  1, 3, 11, 42, 117,  185,  127;
  1, 3, 12, 55, 199,  459,  550,  255;
  1, 3, 13, 70, 315,  981, 1825, 1644,  511;
  1, 3, 14, 87, 471, 1871, 4888, 7287, 4925, 1023;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Rows are A000225, A047926, A073724, A094195, A094259.

A094250:= func< n,k | ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2 >;
[A094250(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Aug 18 2023

A094250[n_, k_]:= ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2;
Table[A094250[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 18 2023 *)

def A094250(n, k): return ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2
flatten([[A094250(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Aug 18 2023

A(n, k) = ((n+2)^(k+1) + (k+1)*n*(n+1) - 1)/(n+1)^2 (array).
T(n, k) = ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2 (antidiagonals).
G.f. for row n: (1-(n+1)*x)/((1-(n+2)*x)*(1-x)^2).

A094259 Expansion of g.f.: (1-5x)/((1-6x)*(1-x)^2).

Original entry on oeis.org

1, 3, 11, 55, 315, 1871, 11203, 67191, 403115, 2418655, 14511891, 87071303, 522427771, 3134566575, 18807399395, 112844396311, 677066377803, 4062398266751, 24374389600435, 146246337602535, 877478025615131, 5264868153690703, 31589208922144131, 189535253532864695

Offset: 0

N. J. A. Sloane, Jun 02 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-13,6).

Cf. A094195.

A row of A094250.

[(6^(n+1) +20*n +19)/25: n in [0..40]]; // G. C. Greubel, Aug 18 2023

LinearRecurrence[{8,-13,6}, {1,3,11}, 41] (* G. C. Greubel, Aug 18 2023 *)

[(6^(n+1) +20*n +19)/25 for n in range(41)] # G. C. Greubel, Aug 18 2023

a(n) = (6^(n+1) + 20*n + 19)/25.

E.g.f.: (1/25)*(6*exp(6*x) + (19 + 20*x)*exp(x)). - G. C. Greubel, Aug 18 2023

A268414 a(n) = 5a(n-1) - 2n for n > 0, a(0) = 1.

Original entry on oeis.org

1, 3, 11, 49, 237, 1175, 5863, 29301, 146489, 732427, 3662115, 18310553, 91552741, 457763679, 2288818367, 11444091805, 57220458993, 286102294931, 1430511474619, 7152557373057, 35762786865245, 178813934326183, 894069671630871, 4470348358154309, 22351741790771497, 111758708953857435

Offset: 0

Ilya Gutkovskiy, Feb 04 2016

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*n, with n > 0 and b(0)=1, is (1 - (m + 2)*x + x^2)/((1 - x)^2*(1 - k*x)). This recurrence gives the closed form b(n) = ((k^2 - k*(m + 2) + 1)*k^n + m*((k - 1)*n + k))/(k - 1)^2.

Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A014827, A024050, A094195, A104745, A107585, A164045, A176916, A221907.

[(4*n + 3*5^n + 5)/8: n in [0..30]]; // Vincenzo Librandi, Feb 06 2016

Table[(4 n + 3 5^n + 5)/8, {n, 0, 23}]
LinearRecurrence[{7, -11, 5}, {1, 3, 11}, 24]

Vec((1-4*x+x^2)/((1-x)^2*(1-5*x)) + O(x^100)) \\ Altug Alkan, Feb 04 2016

G.f.: (1 - 4*x + x^2)/((1 - x)^2*(1 - 5*x)).
a(n) = (4*n + 3*5^n + 5)/8.
Sum_{n>=0} 1/a(n) = 1.449934283402232875...
Lim_{n -> oo} a(n + 1)/a(n) = 5.
From Elmo R. Oliveira, Sep 10 2024: (Start)
E.g.f.: exp(x)*(3*exp(4*x) + 4*x + 5)/8.
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3) for n > 2. (End)

a(24)-a(25) from Elmo R. Oliveira, Sep 10 2024

cp's OEIS Frontend

A094250 Array, A(n, k) = ((n+2)^(k+1) + (k+1)n(n+1) - 1)/(n+1)^2, read by antidiagonals.

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A094259 Expansion of g.f.: (1-5x)/((1-6x)*(1-x)^2).

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A268414 a(n) = 5a(n-1) - 2n for n > 0, a(0) = 1.

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cp's OEIS Frontend

A094250 Array, A(n, k) = ((n+2)^(k+1) + (k+1)*n*(n+1) - 1)/(n+1)^2, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A094259 Expansion of g.f.: (1-5*x)/((1-6*x)*(1-x)^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A268414 a(n) = 5*a(n-1) - 2*n for n > 0, a(0) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A094250 Array, A(n, k) = ((n+2)^(k+1) + (k+1)n(n+1) - 1)/(n+1)^2, read by antidiagonals.

A094259 Expansion of g.f.: (1-5x)/((1-6x)*(1-x)^2).

A268414 a(n) = 5a(n-1) - 2n for n > 0, a(0) = 1.