A015509 -id:A015509

A015502

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

+10
11

1, 1, 5, 70, 2870, 350140, 127801100, 139814403400, 458731057555400, 4514831068460246800, 133300387296288786770000, 11806948504381482999365980000, 3137354163532752044074527571580000, 2500979519710095684958538548015855960000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..50

FORMULA

a(n) = ((3^(n-1) + 1)/2) * a(n-1). - Vincenzo Librandi, Nov 11 2012

a(n) ~ c * 3^(n*(n-1)/2) / 2^(n+1), where c = A132323 = QPochhammer(-1, 1/3) = 3.129868... . - Vaclav Kotesovec, Mar 24 2017

MATHEMATICA

Flatten[{1, Table[QPochhammer[-1, 3, n]/2^(n+1), {n, 2, 15}]}] (* Vaclav Kotesovec, Mar 24 2017 *)

a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1)+m-2)*a[n-1, m]/(m-1)];

Table[a[n, 3], {n, 20}] (* G. C. Greubel, Apr 29 2023 *)

PROG

(Magma) [n le 2 select 1 else ((3^(n-1)+1)/2)*Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 11 2012

(SageMath)

@CachedFunction # a = A015502

def a(n, m): return 1 if (n<3) else (m^(n-1) + m-2)*a(n-1, m)/(m-1)

[a(n, 3) for n in range(1, 31)] # G. C. Greubel, Apr 29 2023

CROSSREFS

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), this sequence (m=3), A015503 (m=4), A015506 (m=5), A015507 (m=6), A015508 (m=7), A015509 (m=8), A015511 (m=9), A015512 (m=10), A015513 (m=11), A015515 (m=12).

Cf. A156296.

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

A015503

a(1) = 1, a(n) = Sum_{k=1..n-1} ((4^k - 1)/3)*a(k).

+10
10

1, 1, 6, 132, 11352, 3882384, 5303336544, 28966824203328, 632809241545903488, 55296137144764138588416, 19327437631660830304254690816, 27021729207700270170039091739231232, 151116480551518237100547636877027177224192

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..50

FORMULA

a(n) = ((4^(n-1) + 2)/3) * a(n-1). - Vincenzo Librandi, Nov 11 2012

MATHEMATICA

a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1)+m-2)*a[n-1, m]/(m-1)];

Table[a[n, 4], {n, 30}] (* G. C. Greubel, Apr 29 2023 *)

PROG

(Magma) [n le 2 select 1 else ((4^(n-1)+2)/3)*Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 11 2012

(SageMath)

@CachedFunction # a = A015503

def a(n, m): return 1 if (n<3) else (m^(n-1) + m-2)*a(n-1, m)/(m-1)

[a(n, 4) for n in range(1, 31)] # G. C. Greubel, Apr 29 2023

CROSSREFS

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), this sequence (m=4), A015506 (m=5), A015507 (m=6), A015508 (m=7), A015509 (m=8), A015511 (m=9), A015512 (m=10), A015513 (m=11), A015515 (m=12).

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

A015506

a(1) = 1, a(n) = Sum_{k=1}^{n-1} (5^k - 1)/4 a(k).

+10
10

1, 1, 7, 224, 35168, 27501376, 107447876032, 2098671914657024, 204950003169660992768, 100073397447688408870744576, 244319893042568615235897903058432, 2982420752607212448380293251367177293824

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..50

FORMULA

a(n) = ((5^(n-1) + 3)/4) * a(n-1). - Vincenzo Librandi, Nov 12 2012

MATHEMATICA

a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1)+m-2)*a[n-1, m]/(m-1)];

Table[a[n, 5], {n, 20}] (* G. C. Greubel, Apr 29 2023 *)

PROG

(Magma) [n le 2 select 1 else ((5^(n-1)+3)/4)*Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012

(SageMath)

@CachedFunction # a = A015506

def a(n, m): return 1 if (n<3) else (m^(n-1) + m-2)*a(n-1, m)/(m-1)

[a(n, 5) for n in range(1, 31)] # G. C. Greubel, Apr 29 2023

CROSSREFS

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), this sequence (m=5), A015507 (m=6), A015508 (m=7), A015509 (m=8), A015511 (m=9), A015512 (m=10), A015513 (m=11), A015515 (m=12).

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

A015507

a(1) = 1, a(n) = Sum_{k=1..n-1} ((6^k - 1)/5)*a(k).

+10
10

1, 1, 8, 352, 91520, 142405120, 1328924579840, 74403829376081920, 24994031979330942894080, 50376471215620688640734003200, 609214555257707874214915513922355200, 44204249911340791820804231319883906967142400

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..50

FORMULA

a(n) = ((6^(n-1) + 4)/5) * a(n-1). - Vincenzo Librandi, Nov 12 2012

MATHEMATICA

a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1)+m-2)*a[n-1, m]/(m-1)];

Table[a[n, 6], {n, 20}] (* G. C. Greubel, Apr 29 2023 *)

PROG

(Magma) [n le 2 select 1 else ((6^(n-1)+4)/5)*Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012

(SageMath)

@CachedFunction

def A015507(n): return 1 if (n<3) else (6^(n-1)+4)*A015507(n-1)/5

[A015507(n) for n in range(1, 21)] # G. C. Greubel, Apr 29 2023

CROSSREFS

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), this sequence (m=6), A015508 (m=7), A015509 (m=8), A015511 (m=9), A015512 (m=10), A015513 (m=11), A015515 (m=12).

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

A015508

a(1) = 1, a(n) = Sum_{k=1..n-1} ((7^k - 1)/6)*a(k).

+10
10

1, 1, 9, 522, 209322, 586520244, 11501075464596, 1578614616119517768, 1516734501782248791012168, 10200952598655696033329019125136, 480252779391204632593567857157274897424, 158269444415262012661462389451687149577571916192

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..49

FORMULA

a(n) = ((7^(n-1) + 5)/6) * a(n-1). - Vincenzo Librandi, Nov 12 2012

MATHEMATICA

a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1) + m-2)*a[n-1, m]/(m-1)];

Table[a[n, 7], {n, 30}] (* G. C. Greubel, Apr 30 2023 *)

PROG

(Magma) [n le 2 select 1 else ((7^(n-1) + 5)/6)*Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012

(SageMath)

@CachedFunction # a = A015508

def a(n, m): return 1 if (n<3) else (m^(n-1)+m-2)*a(n-1, m)/(m-1)

[a(n, 7) for n in range(1, 31)] # G. C. Greubel, Apr 30 2023

CROSSREFS

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), A015507 (m=6), this sequence (m=7), A015509 (m=8), A015511 (m=9), A015512 (m=10), A015513 (m=11), A015515 (m=12).

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

A015511

a(1) = 1, a(n) = Sum_{k=1..n-1} ((9^k - 1)/8)*a(k).

+10
10

1, 1, 11, 1012, 830852, 6133349464, 407444538242984, 243599680968409330048, 1310771150941736627904810368, 63477451180042308935531134194562816, 27666523379269090447091129488519658150671616

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..47

FORMULA

a(n) = ((9^(n-1) + 7)/8) * a(n-1). - Vincenzo Librandi, Nov 12 2012

a(n) ~ QPochhammer(-63, 1/9) * 3^(n*(n-1)) / 2^(3*n+7). - Vaclav Kotesovec, May 03 2023

MATHEMATICA

a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1) +m-2)*a[n-1, m]/(m-1)];

Table[a[n, 9], {n, 30}] (* G. C. Greubel, May 03 2023 *)

Join[{1}, Table[7^n*QPochhammer[-1/7, 9, n]/2^(3*n + 1), {n, 2, 12}]] (* Vaclav Kotesovec, May 03 2023 *)

PROG

(Magma) [n le 2 select 1 else ((9^(n-1)+7)/8)*Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012

(SageMath)

@CachedFunction # a = A015511

def a(n, m): return 1 if (n<3) else (m^(n-1)+m-2)*a(n-1, m)/(m-1)

[a(n, 9) for n in range(1, 31)] # G. C. Greubel, May 03 2023

CROSSREFS

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), A015507 (m=6), A015508 (m=7), A015509 (m=8), this sequence (m=9), A015512 (m=10), A015513 (m=11), A015515 (m=12).

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

A015512

a(1) = 1, a(n) = Sum_{k=1..n-1} ((10^k - 1)/9)*a(k).

+10
10

1, 1, 12, 1344, 1494528, 16607195136, 1845258665951232, 2050289046842405289984, 22780991231839211526404702208, 2531221268231904597902043824359735296, 2812468078063201791652852780757078172764209152

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..46

FORMULA

a(n) = ((10^(n-1) + 8)/9) * a(n-1). - Vincenzo Librandi, Nov 12 2012

MATHEMATICA

a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1) +m-2)*a[n-1, m]/(m-1)];

Table[a[n, 10], {n, 30}] (* G. C. Greubel, May 03 2023 *)

PROG

(Magma) [n le 2 select 1 else ((10^(n-1) + 8)/9)*Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012

(SageMath)

@CachedFunction # a = A015512

def a(n, m): return 1 if (n<3) else (m^(n-1)+m-2)*a(n-1, m)/(m-1)

[a(n, 10) for n in range(1, 31)] # G. C. Greubel, May 03 2023

CROSSREFS

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), A015507 (m=6), A015508 (m=7), A015509 (m=8), A015511 (m=9), this sequence (m=10), A015513 (m=11), A015515 (m=12).

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

A015513

a(1) = 1, a(n) = Sum_{k=1..n-1} ((11^k - 1)/10)*a(k).

+10
10

1, 1, 13, 1742, 2552030, 41102995180, 7281683317103260, 14189947350338830620680, 304174136317707285574697584520, 71722670512982436329410134761448960400, 186030135925835196854820049614502274473787544400

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..45

FORMULA

a(n) = ((11^(n-1) + 9)/10) * a(n-1). - Vincenzo Librandi, Nov 12 2012

MATHEMATICA

a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1) +m-2)*a[n-1, m]/(m-1)];

Table[a[n, 10], {n, 30}] (* G. C. Greubel, May 03 2023 *)

PROG

(Magma) [n le 2 select 1 else ((11^(n-1) + 9)/10) * Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012

(SageMath)

def a(n, m) -> int: # a = A015513

return 1 if (n<3) else (m^(n-1)+m-2)*a(n-1, m)//(m-1)

[a(n, 11) for n in range(1, 31)] # G. C. Greubel, May 03 2023

CROSSREFS

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), A015507 (m=6), A015508 (m=7), A015509 (m=8), A015511 (m=9), A015512 (m=10), this sequence (m=11), A015515 (m=12).

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

A015515

a(1) = 1, a(n) = Sum_{k=1..n-1} ((12^k - 1)/11)*a(k).

+10
10

1, 1, 14, 2212, 4171832, 94375183504, 25618521062894816, 83450744014073963641408, 3262026661649164626974053098368, 1530121919008888925087797696409496422656, 8612828743790947623482719127044813092555596516864

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..44

FORMULA

a(n) = ((12^(n-1) + 10)/11) * a(n-1). - Vincenzo Librandi, Nov 12 2012

MATHEMATICA

Join[{1}, RecurrenceTable[{a[2]==1, a[n]==(12^(n-1)+10)/11 a[n-1]}, a, {n, 12}]] (* Harvey P. Dale, Mar 10 2013 *)

PROG

(Magma) [n le 2 select 1 else ((12^(n-1) + 10)/11) * Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012

(SageMath)

def a(n, m) -> int: # a = A015515

return 1 if (n<3) else (m^(n-1)+m-2)*a(n-1, m)//(m-1)

[a(n, 12) for n in range(1, 31)] # G. C. Greubel, May 03 2023

CROSSREFS

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), A015507 (m=6), A015508 (m=7), A015509 (m=8), A015511 (m=9), A015512 (m=10), A015513 (m=11), this sequence (m=12).

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved