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.

Previous Showing 51-60 of 88 results. Next

A213546 Principal diagonal of the convolution array A213505.

Original entry on oeis.org

1, 25, 170, 674, 1979, 4795, 10164, 19524, 34773, 58333, 93214, 143078, 212303, 306047, 430312, 592008, 799017, 1060257, 1385746, 1786666, 2275427, 2865731, 3572636, 4412620, 5403645, 6565221, 7918470, 9486190, 11292919, 13364999, 15730640, 18419984, 21465169
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2012

Keywords

Links

Crossrefs

Cf. A213505, A213500.

Programs

  • Mathematica
    (See A213505.)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,25,170,674,1979,4795},40] (* Harvey P. Dale, May 12 2025 *)

Formula

a(n) = (16*n^5 + 15*n^4 - n)/30.
a(n) = 6*a(n-1) - 10*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: x*(1 + 19*x + 35*x^2 + 9*x^3)/(1 - x)^6.
E.g.f.: exp(x)*x*(30 + 345*x + 490*x^2 + 175*x^3 + 16*x^4)/30. - Stefano Spezia, Oct 28 2023

A213549 Principal diagonal of the convolution array A213548.

Original entry on oeis.org

1, 12, 53, 155, 360, 721, 1302, 2178, 3435, 5170, 7491, 10517, 14378, 19215, 25180, 32436, 41157, 51528, 63745, 78015, 94556, 113597, 135378, 160150, 188175, 219726, 255087, 294553, 338430, 387035, 440696, 499752, 564553, 635460
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2012

Keywords

Links

Crossrefs

Cf. A000217, A213500, A213548.

Programs

  • Mathematica
    (See A213548.)
LinearRecurrence[{5,-10,10,-5,1},{1,12,53,155,360},40] (* Harvey P. Dale, Nov 20 2022 *)

Formula

G.f.: x*(1 + 7*x + 3*x^2)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = (11*n^4 + 14*n^3 + n^2 - 2*n)/24 = n*(n+1)*(11*n^2 + 3*n - 2)/24.
a(n) = Sum_{i=1..n} (n-i+1)*A000217(n+i-1), see A213548. - Bruno Berselli, Oct 05 2016

A213552 Principal diagonal of the convolution array A213551.

Original entry on oeis.org

1, 15, 81, 281, 756, 1722, 3486, 6462, 11187, 18337, 28743, 43407, 63518, 90468, 125868, 171564, 229653, 302499, 392749, 503349, 637560, 798974, 991530, 1219530, 1487655, 1800981, 2164995, 2585611, 3069186, 3622536, 4252952
Offset: 1

Views

Author

Clark Kimberling, Jun 17 2012

Keywords

Links

Crossrefs

Cf. A213551, A213500.

Programs

Formula

a(n) = n*(1 + n)*(2 + n)(-3 + 7*n + 16*n^2)/120.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: x*(1 + 9*x + 6*x^2)/(1 - x)^6.

A213554 Principal diagonal of the convolution array A213553.

Original entry on oeis.org

1, 43, 334, 1406, 4271, 10577, 22764, 44220, 79437, 134167, 215578, 332410, 495131, 716093, 1009688, 1392504, 1883481, 2504067, 3278374, 4233334, 5398855, 6807977, 8497028, 10505780, 12877605, 15659631, 18902898, 22662514
Offset: 1

Views

Author

Clark Kimberling, Jun 17 2012

Keywords

Links

Crossrefs

Cf. A213500, A213553.

Programs

  • GAP
    List([1..30], n-> n*(39*n^4 +15*n^3 -25*n^2 +1)/30); # G. C. Greubel, Jul 31 2019
  • Magma
    [n*(39*n^4 +15*n^3 -25*n^2 +1)/30: n in [1..30]]; // G. C. Greubel, Jul 31 2019
  • Mathematica
    (* First program *)
b[n_]:= n; 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}]  (* A213553 *)
d = Table[T[n, n], {n, 1, 40}] (* A213554 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A101089 *)
(* Second program *)
Table[(39n^5+15n^4-25n^3+n)/30,{n,30}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{1,43,334,1406,4271,10577},30] (* Harvey P. Dale, Jan 15 2013 *)
  • PARI
    vector(30, n, n*(39*n^4 +15*n^3 -25*n^2 +1)/30) \\ G. C. Greubel, Jul 31 2019
  • Sage
    [n*(39*n^4 +15*n^3 -25*n^2 +1)/30 for n in (1..30)] # G. C. Greubel, Jul 31 2019

Formula

a(n) = n*(39*n^4 + 15*n^3 - 25*n^2 + 1)/30.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: x*(1 + 37*x + 91*x^2 + 27*x^3)/(1 - x)^6.

A213556 Principal diagonal of the convolution array A213555.

Original entry on oeis.org

1, 19, 118, 446, 1271, 3017, 6300, 11964, 21117, 35167, 55858, 85306, 126035, 181013, 253688, 348024, 468537, 620331, 809134, 1041334, 1324015, 1664993, 2072852, 2556980, 3127605, 3795831, 4573674, 5474098, 6511051, 7699501
Offset: 1

Views

Author

Clark Kimberling, Jun 17 2012

Keywords

Links

Crossrefs

Cf. A213553, A213500.

Programs

Formula

a(n) = (9*n^5 + 15*n^4 + 5*n^3 + n)/30.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: x*(1 + 13*x + 19*x^2 + 3*x^3)/(1 - x)^6.

A213559 Principal diagonal of the convolution array A213558.

Original entry on oeis.org

1, 91, 1366, 9542, 43535, 151313, 435324, 1089804, 2452269, 5071495, 9794290, 17873362, 31098587, 51953981, 83802680, 131102232, 199652505, 296878515, 432150478, 617143390, 866238439, 1196968553, 1630510388, 2192225060
Offset: 1

Views

Author

Clark Kimberling, Jun 17 2012

Keywords

Links

Crossrefs

Cf. A213558, A213500.

Programs

  • Mathematica
    (See A213558.)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,91,1366,9542,43535,151313,435324,1089804},40] (* Harvey P. Dale, Oct 09 2016 *)

Formula

a(n) = (16/35)*n^7 + (1/2)*n^6 + (1/15)*n^3 - (1/42)*n.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
G.f.: x*(1 + 83*x + 666*x^2 + 1106*x^3 + 421*x^4 + 27*x^5)/(1 - x)^8

A213560 Antidiagonal sums of the convolution array A213558.

Original entry on oeis.org

1, 24, 236, 1400, 6009, 20608, 59952, 153792, 357225, 765688, 1535820, 2913560, 5270993, 9153600, 15339712, 24914112, 39357873, 60656664, 91429900, 135083256, 195987209, 279684416, 393128880, 544960000, 745814745, 1008681336, 1349297964, 1786600216, 2343221025
Offset: 1

Views

Author

Clark Kimberling, Jun 17 2012

Keywords

Comments

An m-star is an m-antichain with a smallest element adjoined. Then, a(n) is the number of proper mergings of a 3-star and an (n-1)-chain. - Henri Mühle, Jan 23 2013
Convolution of A000578 and A000583. - Stefano Spezia, Apr 07 2023

Links

Crossrefs

Cf. A000578, A000583, A213558, A213500.

Programs

Formula

a(n) = n*(1 + n)^2*(2 + n)*(16 + 18*n + 21*n^2 + 12*n^3 + 3*n^4)/840.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).
G.f.: x*(1 + x)*(1 + 4*x + x^2)*(1 + 10*x + x^2)/(1 - x)^9.

A213562 Principal diagonal of the convolution array A213561.

Original entry on oeis.org

1, 18, 109, 407, 1152, 2723, 5670, 10746, 18939, 31504, 49995, 76297, 112658, 161721, 226556, 310692, 418149, 553470, 721753, 928683, 1180564, 1484351, 1847682, 2278910, 2787135, 3382236, 4074903, 4876669, 5799942, 6858037
Offset: 1

Views

Author

Clark Kimberling, Jun 18 2012

Keywords

Links

Crossrefs

Cf. A213561, A213500.

Programs

  • Mathematica
    (See A213561.)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,18,109,407,1152,2723},30] (* Harvey P. Dale, Nov 12 2014 *)

Formula

a(n) = (4/15)*n^5 + (11/24)*n^4 + (1/4)*n^3 + (1/24)*n^2 - (1/16)*n.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: x*(1 + 12*x + 16*x^2 + 3*x^3)/(1 - x)^6.

A213563 Antidiagonal sums of the convolution array A213561.

Original entry on oeis.org

1, 10, 51, 182, 518, 1260, 2730, 5412, 9999, 17446, 29029, 46410, 71708, 107576, 157284, 224808, 314925, 433314, 586663, 782782, 1030722, 1340900, 1725230, 2197260, 2772315, 3467646, 4302585, 5298706, 6479992, 7873008, 9507080
Offset: 1

Views

Author

Clark Kimberling, Jun 18 2012

Keywords

Links

Crossrefs

Cf. A213561, A213500.

Programs

  • Mathematica
    (See A213561.)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,10,51,182,518,1260,2730},40] (* Harvey P. Dale, Aug 10 2024 *)

Formula

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: x*(1 + 3 x + 2*x^2)/(1 - x)^7.

A213565 Principal diagonal of the convolution array A213564.

Original entry on oeis.org

1, 21, 127, 467, 1302, 3038, 6258, 11754, 20559, 33979, 53625, 81445, 119756, 171276, 239156, 327012, 438957, 579633, 754243, 968583, 1229074, 1542794, 1917510, 2361710, 2884635, 3496311, 4207581, 5030137, 5976552, 7060312
Offset: 1

Views

Author

Clark Kimberling, Jun 18 2012

Keywords

Links

Crossrefs

Cf. A213564, A213500.

Programs

Formula

a(n) = (16*n^5 + 85*n^4 + 15*n^3 - 25*n^2 - n)/60.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: x*(1 + 15*x + 16*x^2)/(1 - x)^6.
Previous Showing 51-60 of 88 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.