A067410 -id:A067410 - cp's OEIS frontend

A067411 Third column of triangle A067410 and second column of A067417.

1, 4, 24, 144, 864, 5184, 31104, 186624, 1119744, 6718464, 40310784, 241864704, 1451188224, 8707129344, 52242776064, 313456656384, 1880739938304, 11284439629824, 67706637778944, 406239826673664

Offset: 0

Wolfdieter Lang, Jan 25 2002

Let f(k) be the sum of the smallest three positive divisors of k, g(k) be the sum of the largest two positive divisors of k, this sequence from a(2) onwards contains the numbers k for which g(k) is a positive integer power of f(k). - Yifan Xie, Jan 27 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, Example 14.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Craig Knecht, Number of tilings for a stackable six sphinx tile shape.
Index entries for linear recurrences with constant coefficients, signature (6).

A002001, A067412 (second and fourth column of A067410), A000244, A067403 (first and third column of A067417), A000400 (powers of 6).

Row sums of A038195.

CoefficientList[Series[(1-2x)/(1-6x),{x,0,30}],x] (* Harvey P. Dale, Feb 26 2015 *)

a(n) = if(n<=0, 0, 4*6^(n-1) ); \\ Joerg Arndt, Feb 23 2014

a(n) = A067410(n+2, 2) = A067417(n+1, 1).
a(n) = 4 * 6^(n-1), for n >= 1, a(0)=1.
G.f.: (1-2*x)/(1-6*x).
E.g.f.: (2*exp(6*x)+1) / 3 = exp(3*x)*(cosh(3*x) + sinh(3*x)/3). - Paul Barry, Nov 20 2003
a(n) = Sum_{k=0..n} C(n,k) * A001045(n+k+1). - Paul Barry, Apr 19 2010

Incorrect formula deleted by Harvey P. Dale, Feb 26 2015

Formula restored by Sean A. Irvine, Jan 10 2021

A067412 Fourth column of triangle A067410.

Original entry on oeis.org

1, 5, 40, 320, 2560, 20480, 163840, 1310720, 10485760, 83886080, 671088640, 5368709120, 42949672960, 343597383680, 2748779069440, 21990232555520, 175921860444160, 1407374883553280, 11258999068426240

Offset: 0

Wolfdieter Lang, Jan 25 2002

The fifth column gives [1,6,60,600,6000,60000,...].

a(n+1) = A157176(A016957(n)). [From Reinhard Zumkeller, Feb 24 2009]

Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A067411 (third column), A067413 (sixth column), A001018 (powers of 8).

Join[{1},NestList[8#&,5,20]] (* or *) CoefficientList[Series[ (1-3x)/ (1-8x),{x,0,20}],x] (* Harvey P. Dale, May 14 2011 *)

a(n)= A067410(n+3, 3). a(n)= 5*8^(n-1), n>=1, a(0)=1.
G.f.: (1-3*x)/(1-8*x).
E.g.f.: (5*exp(8*x)+3)/8 = exp(4*x)*(cosh(4*x)+sinh(4*x)/4) - Paul Barry, Nov 20 2003

A067413 Sixth column of triangle A067410.

Original entry on oeis.org

1, 7, 84, 1008, 12096, 145152, 1741824, 20901888, 250822656, 3009871872, 36118462464, 433421549568, 5201058594816, 62412703137792, 748952437653504, 8987429251842048, 107849151022104576, 1294189812265254912

Offset: 0

Wolfdieter Lang, Jan 25 2002

The fifth column is [1,6,60,600,6000,60000,...].

Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A067412 (fourth column), A067414 (seventh column), A001021 (powers of 12).

a(n)= A067410(n+5, 5). a(n)= 7*12^(n-1), n>=1, a(0)=1.

G.f.: (1-5*x)/(1-12*x).

A067414 Seventh column of triangle A067410.

Original entry on oeis.org

1, 8, 112, 1568, 21952, 307328, 4302592, 60236288, 843308032, 11806312448, 165288374272, 2314037239808, 32396521357312, 453551299002368, 6349718186033152, 88896054604464128, 1244544764462497792

Offset: 0

Wolfdieter Lang, Jan 25 2002

Index entries for linear recurrences with constant coefficients, signature (14).

Cf. A067413 (sixth column), A067415 (eighth column), A001023 (powers of 14).

a(n) = A067410(n+6, 6).
a(n) = 8*14^(n-1), n>=1, a(0)=1.
G.f.: (1-6*x)/(1-14*x).

A067415 Eighth column of triangle A067410.

Original entry on oeis.org

1, 9, 144, 2304, 36864, 589824, 9437184, 150994944, 2415919104, 38654705664, 618475290624, 9895604649984, 158329674399744, 2533274790395904, 40532396646334464, 648518346341351424

Offset: 0

Wolfdieter Lang, Jan 25 2002

Index entries for linear recurrences with constant coefficients, signature (16).

Cf. A067414 (seventh column), A067416 (ninth column), A001025 (powers of 16).

a(n)= A067410(n+7, 7). a(n)= 9*16^(n-1), n>=1, a(0)=1.

G.f.: (1-7*x)/(1-16*x).

A067416 Ninth column of triangle A067410.

Original entry on oeis.org

1, 10, 180, 3240, 58320, 1049760, 18895680, 340122240, 6122200320, 110199605760, 1983592903680, 35704672266240, 642684100792320, 11568313814261760, 208229648656711680, 3748133675820810240

Offset: 0

Wolfdieter Lang, Jan 25 2002

Index entries for linear recurrences with constant coefficients, signature (18).

Cf. A067415 (eighth column), A001027 (powers of 18).

a(n)= A067410(n+8, 8). a(n)= 10*18^(n-1), n>=1, a(0)=1.

G.f. (1-8*x)/(1-18*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

Wolfdieter Lang, Jan 25 2002

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

Indranil Ghosh, Rows 0..125, flattened

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.

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 *)

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).

A067425 Triangle with columns built from certain power sequences.

Original entry on oeis.org

1, 4, 1, 16, 5, 1, 64, 40, 6, 1, 256, 320, 72, 7, 1, 1024, 2560, 864, 112, 8, 1, 4096, 20480, 10368, 1792, 160, 9, 1, 16384, 163840, 124416, 28672, 3200, 216, 10, 1, 65536, 1310720, 1492992, 458752, 64000

Offset: 0

Wolfdieter Lang, Jan 25 2002

The fifth column (m=4) gives [1, 8, 160, 3200, 64000, 1280000, 25600000, ...].

			Triangle starts:
   1;
   4,  1;
  16,  5,  1;
  64, 40,  6,  1;
  ...

Indranil Ghosh, Rows 0..125, flattened

Cf. A009998, A067410, A067417.
Columns 0..3 are A000302 (powers of 4), A067412, A067419, A067404.
Columns 5..8 are A067426, A067427, A067428, A067429.

A067425[n_, m_] := If[n == m, 1, (m + 4)*(4*(m + 1))^(n - m - 1)];
Table[A067425[n, m], {n, 0, 10}, {m, 0, n}] (* Paolo Xausa, Oct 16 2024 *)

T(n,m) = 1 if n = m; T(n,m) = (m+4)*(4*(m+1))^(n-m-1) if n > m >= 0, else 0.

G.f. for column m: (x^m)*(1-3*m*x)/(1-4*(m+1)*x).

A090019 a(n) = (310^n + 20^n)/5.

Original entry on oeis.org

1, 6, 60, 600, 6000, 60000, 600000, 6000000, 60000000, 600000000, 6000000000, 60000000000, 600000000000, 6000000000000, 60000000000000, 600000000000000, 6000000000000000, 60000000000000000, 600000000000000000

Offset: 0

Paul Barry, Nov 20 2003

Fifth column of triangle A067410.

Index entries for linear recurrences with constant coefficients, signature (10).

Join[{1},NestList[10#&,6,20]] (* Harvey P. Dale, Apr 02 2015 *)

G.f.: (1-4x)/(1-10x).

E.g.f.: (3*exp(10x)+2)/5 = exp(5x)(cosh(5x)+sinh(5x)/5).

cp's OEIS Frontend

A067411 Third column of triangle A067410 and second column of A067417.

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A067412 Fourth column of triangle A067410.

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A067413 Sixth column of triangle A067410.

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A067414 Seventh column of triangle A067410.

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A067415 Eighth column of triangle A067410.

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A067416 Ninth column of triangle A067410.

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A067417 Triangle with columns built from certain power sequences.

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A067425 Triangle with columns built from certain power sequences.

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A090019 a(n) = (3*10^n + 2*0^n)/5.

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A090019 a(n) = (310^n + 20^n)/5.