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.

Showing 1-9 of 9 results.

A067403 Third column of triangle A067402.

Original entry on oeis.org

1, 5, 45, 405, 3645, 32805, 295245, 2657205, 23914845, 215233605, 1937102445, 17433922005, 156905298045, 1412147682405, 12709329141645, 114383962274805, 1029455660473245, 9265100944259205, 83385908498332845, 750473176484995605, 6754258588364960445, 60788327295284644005
Offset: 0

Views

Author

Wolfdieter Lang, Jan 25 2002

Keywords

Links

Crossrefs

Cf. A002001 (second column), A067404 (fourth column), A001019 (powers of 9).
Cf. A067402.

Programs

  • Maple
    A067403:=n->5*9^(n-1): 1,seq(A067403(n), n=1..30); # Wesley Ivan Hurt, Apr 09 2017
  • Mathematica
    Join[{1},NestList[9#&,5,30]] (* or *) CoefficientList[Series[ (1-4x)/ (1-9x),{x,0,30}],x] (* Harvey P. Dale, Apr 26 2011 *)
  • PARI
    Vec((1-4*x)/(1-9*x) + O(x^30)) \\ Michel Marcus, Apr 09 2017

Formula

a(n) = A067402(n+2, 2).
a(n) = 5*9^(n-1) for n>=1, a(0) = 1.
G.f.: (1-4*x)/(1-9*x).
E.g.f.: (4 + 5*exp(9*x))/9. - Stefano Spezia, Sep 30 2022

A067404 Fourth column of triangle A067402.

Original entry on oeis.org

1, 7, 112, 1792, 28672, 458752, 7340032, 117440512, 1879048192, 30064771072, 481036337152, 7696581394432, 123145302310912, 1970324836974592, 31525197391593472, 504403158265495552, 8070450532247928832, 129127208515966861312, 2066035336255469780992, 33056565380087516495872
Offset: 0

Views

Author

Wolfdieter Lang, Jan 25 2002

Keywords

Links

Crossrefs

Cf. A067403 (third column), A067405 (fifth column), A001025 (powers of 16).
Cf. A067402.

Programs

  • Mathematica
    CoefficientList[Series[(1-9x)/(1-16x),{x,0,20}],x] (* Harvey P. Dale, Jun 02 2017 *)

Formula

a(n) = A067402(n+3, 3).
a(n) = 7*16^(n-1), n>=1, a(0) = 1.
G.f.: (1-9*x)/(1-16*x).
E.g.f.: (9 + 7*exp(16*x))/16. - Stefano Spezia, Sep 30 2022

Extensions

More terms from Harvey P. Dale, Jun 02 2017

A067405 Fifth column of triangle A067402.

Original entry on oeis.org

1, 9, 225, 5625, 140625, 3515625, 87890625, 2197265625, 54931640625, 1373291015625, 34332275390625, 858306884765625, 21457672119140625, 536441802978515625, 13411045074462890625, 335276126861572265625, 8381903171539306640625, 209547579288482666015625, 5238689482212066650390625
Offset: 0

Views

Author

Wolfdieter Lang, Jan 25 2002

Keywords

Links

Crossrefs

Cf. A067404 (fourth column), A067406 (sixth column), A009969 (powers of 25).
Cf. A067402.

Programs

  • Mathematica
    Join[{1},NestList[25#&,9,20]] (* Harvey P. Dale, Mar 12 2023 *)

Formula

a(n) = A067402(n+4, 4).
a(n) = 9*25^(n-1), n>=1, a(0) = 1.
G.f.: (1-16*x)/(1-25*x).
E.g.f.: (16 + 9*exp(25*x))/25. - Stefano Spezia, Sep 30 2022

A067406 Sixth column of triangle A067402.

Original entry on oeis.org

1, 11, 396, 14256, 513216, 18475776, 665127936, 23944605696, 862005805056, 31032208982016, 1117159523352576, 40217742840692736, 1447838742264938496, 52122194721537785856, 1876399009975360290816, 67550364359112970469376, 2431813116928066936897536, 87545272209410409728311296
Offset: 0

Views

Author

Wolfdieter Lang, Jan 25 2002

Keywords

Links

Crossrefs

Cf. A067405 (fifth column), A067407 (seventh column), A009980 (powers of 36).
Cf. A067402.

Formula

a(n) = A067402(n+5, 5).
a(n) = 11*36^(n-1), n>=1, a(0) = 1.
G.f.: (1-25*x)/(1-36*x).
E.g.f.: (25 + 11*exp(36*x))/36. - Stefano Spezia, Sep 30 2022

A067407 Seventh column of triangle A067402.

Original entry on oeis.org

1, 13, 637, 31213, 1529437, 74942413, 3672178237, 179936733613, 8816899947037, 432028097404813, 21169376772835837, 1037299461868956013, 50827673631578844637, 2490556007947363387213, 122037244389420805973437, 5979824975081619492698413, 293011423778999355142222237
Offset: 0

Views

Author

Wolfdieter Lang, Jan 25 2002

Keywords

Links

Crossrefs

Cf. A067406 (sixth column), A067408 (eighth column).
Cf. A067402.

Formula

a(n) = A067402(n+6, 6).
a(n) = 13*49^(n-1), n>=1, a(0) = 1.
G.f.: (1-36*x)/(1-49*x).
E.g.f.: (36 + 13*exp(49*x))/49. - Stefano Spezia, Sep 30 2022

A067408 Eighth column of triangle A067402.

Original entry on oeis.org

1, 15, 960, 61440, 3932160, 251658240, 16106127360, 1030792151040, 65970697666560, 4222124650659840, 270215977642229760, 17293822569102704640, 1106804644422573096960, 70835497243044678205440, 4533471823554859405148160, 290142196707511001929482240, 18569100589280704123486863360
Offset: 0

Views

Author

Wolfdieter Lang, Jan 25 2002

Keywords

Links

Crossrefs

Cf. A067407 (seventh column), A067409 (ninth column).
Cf. A067402.

Programs

  • Mathematica
    LinearRecurrence[{64},{1,15},20] (* or *) Join[{1},NestList[64#&,15,20]] (* Harvey P. Dale, Aug 25 2016 *)

Formula

a(n) = A067402(n+7, 7).
G.f.: (1-49*x)/(1-64*x).
E.g.f.: (49 + 15*exp(64*x))/64. - Stefano Spezia, Sep 30 2022

Extensions

Incorrect formula deleted by Harvey P. Dale, Aug 25 2016

A067409 Ninth column of triangle A067402.

Original entry on oeis.org

1, 17, 1377, 111537, 9034497, 731794257, 59275334817, 4801302120177, 388905471734337, 31501343210481297, 2551608800048985057, 206680312803967789617, 16741105337121390958977, 1356029532306832667677137, 109838392116853446081848097, 8896909761465129132629695857
Offset: 0

Views

Author

Wolfdieter Lang, Jan 25 2002

Keywords

Comments

The tenth column gives [1, 19, 1900, 190000, ...].

Links

Crossrefs

Cf. A067408 (eighth column).
Cf. A067402.

Programs

  • Mathematica
    Join[{1},NestList[81#&,17,15]] (* Harvey P. Dale, Nov 19 2024 *)

Formula

a(n) = A067402(n+8, 8).
a(n) = 17*81^(n-1), n>=1, a(0) = 1.
G.f.: (1-64*x)/(1-81*x).
E.g.f.: (64 + 17*exp(81*x))/81. - Stefano Spezia, Sep 30 2022

A067410 Triangle with columns built from certain power sequences.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 12, 4, 1, 16, 48, 24, 5, 1, 32, 192, 144, 40, 6, 1, 64, 768, 864, 320, 60, 7, 1, 128, 3072, 5184, 2560, 600, 84, 8, 1, 256, 12288, 31104, 20480, 6000, 1008, 112, 9, 1, 512, 49152, 186624, 163840, 60000, 12096, 1568, 144, 10, 1
Offset: 0

Views

Author

Wolfdieter Lang, Jan 25 2002

Keywords

Examples

			Triangle starts:
  1;
  2,  1;
  4,  3, 1;
  8, 12, 4, 1;
  ...

Links

Crossrefs

Cf. A009998 (triangle built from powers of (m+1)), A067402.
Columns m=0..8 are A000079, A002001, A067411, A067412, A090019, A067413, A067414, A067415, A067416.

Programs

  • Mathematica
    A[n_,m_]:=If[n==m,1,(m+2)(2(m+1))^(n-m-1)]; Flatten[Table[A[n,m],{n,0,9},{m,0,n}]] (* Stefano Spezia, Sep 30 2022 *)

Formula

a(n, m) = 1 if n = m; a(n, m) = (m+2)*(2*(m+1))^(n-m-1) if n > m >= 0.
G.f. for column m: (x^m)*(1-m*x)/(1-2*(m+1)*x).

A067417 Triangle with columns built from certain power sequences.

Original entry on oeis.org

1, 3, 1, 9, 4, 1, 27, 24, 5, 1, 81, 144, 45, 6, 1, 243, 864, 405, 72, 7, 1, 729, 5184, 3645, 864, 105, 8, 1, 2187, 31104, 32805, 10368, 1575, 144, 9, 1, 6561, 186624, 295245, 124416, 23625, 2592, 189, 10, 1, 19683, 1119744, 2657205, 1492992, 354375, 46656, 3969, 240, 11, 1
Offset: 0

Views

Author

Wolfdieter Lang, Jan 25 2002

Keywords

Examples

			Triangle starts:
   1;
   3,  1;
   9,  4, 1;
  27, 24, 5, 1;
  ...

Links

Crossrefs

Cf. A009998 (triangle built from powers of (m+1)), A067402, A067410.
Columns m=0..8 are A000244, A067411, A067403, A067419, A067420, A067421, A067422, A067423, A067424.

Programs

  • Mathematica
    A[n_,m_]:=If[n==m,1,(m+3)(3(m+1))^(n-m-1)]; Flatten[Table[A[n,m],{n,0,9},{m,0,n}]] (* Stefano Spezia, Sep 30 2022 *)

Formula

a(n, m) = 1 if n = m; a(n, m) = (m+3)*(3*(m+1))^(n-m-1) if n > m >= 0.
G.f. for column m: (x^m)*(1-2*m*x)/(1-3*(m+1)*x).
Showing 1-9 of 9 results.

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

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