A014480 -id:A014480 - cp's OEIS frontend

A167581 The second left hand column of triangle A167580.

-1, 0, 28, 192, 880, 3328, 11200, 34816, 102144, 286720, 777216, 2048000, 5271552, 13303808, 33013760, 80740352, 194969600, 465567744, 1100742656, 2579496960, 5996806144, 13841203200, 31738298368, 72343355392, 163997286400

Offset: 2

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (8, -24, 32, -16).

Equals the second left hand column of triangle A167580.

Other left hand columns are A014480 A167582, A168305 and A168306.

LinearRecurrence[{8,-24,32,-16},{-1,0,28,192},30] (* Harvey P. Dale, Jan 22 2012 *)

a(n) = 2^n*(2*n^3 - 9*n^2 + 10*n - 3)/12.
GF(z) = (4*z^2 + 8*z - 1)/(1-2*z)^4.
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4)
a(n) - 7*a(n-1) + 18*a(n-2) - 20*a(n-3) + 8*a(n-4) = 1*2^(n-1)

Formulae and links added by Johannes W. Meijer, Nov 23 2009

A167582 The third left hand column of triangle A167580.

Original entry on oeis.org

3, 98, 1080, 7568, 40976, 187488, 761600, 2830848, 9821952, 32254464, 101263360, 306229248, 897175552, 2558058496, 7123894272, 19434569728, 52063567872, 137236709376, 356550967296, 914355126272, 2317328842752

Offset: 3

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (12, -60, 160, -240, 192, -64).

Equals the third left hand column of triangle A167580.

Other left hand columns are A014480, A167581, A168305 and A168306.

LinearRecurrence[{12, -60, 160, -240, 192, -64}, {3, 98, 1080, 7568, 40976, 187488}, 100] (* G. C. Greubel, Jun 16 2016 *)

a(n) = 2^n*(14*n^5 - 95*n^4 + 240*n^3 - 280*n^2 + 151*n - 30)/240.
GF(z) = (8*z^3 + 84*z^2 + 62*z + 3)/(1-2*z)^6.
a(n) = 12*a(n-1) - 60*a(n-2) + 160*a(n-3) - 240*a(n-4) + 192*a(n-5) - 64*a(n-6).
a(n)-11*a(n-1)+50*a(n-2)-120*a(n-3)+160*a(n-4)-112*a(n-5)+32*a(n-6) = 7*2^(n-1).

Formulae and links added by Johannes W. Meijer, Nov 23 2009

A168305 The fourth left hand column of triangle A167580.

Original entry on oeis.org

-15, -48, 2024, 31616, 274480, 1784320, 9645312, 45735936, 196441344, 780595200, 2912532480, 10315202560, 34963222528, 114140905472, 360716042240, 1108051230720, 3319564664832, 9726122262528, 27935264735232, 78810426900480, 218761889054720, 598349308755968

Offset: 4

Johannes W. Meijer, Nov 23 2009

G. C. Greubel, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (16, -112, 448, -1120, 1792, -1792, 1024, -256).

Equals the fourth left hand column of triangle A167580.

Other left hand columns are A014480, A167581, A167582 and A168306.

[2^n*(54*n^7-763*n^6+4158*n^5-11305*n^4+16401*n^3- 12502*n^2+4587*n-630)/10080: n in [4..40]]; // Vincenzo Librandi, Jul 18 2016

LinearRecurrence[{16, -112, 448, -1120, 1792, -1792, 1024, -256}, {-15, -48, 2024, 31616, 274480, 1784320, 9645312, 45735936}, 997] (* G. C. Greubel, Jul 17 2016 *)

a(n) = 2^n*(54*n^7 - 763*n^6 + 4158*n^5 - 11305*n^4 + 16401*n^3 - 12502*n^2 + 4587*n - 630)/10080.
G.f.: (16*z^4 + 576*z^3 + 1112*z^2 + 192*z - 15)/(2*z-1)^8.
a(n) = 16*a(n-1) - 112*a(n-2) + 448*a(n-3) - 1120*a(n-4) + 1792*a(n-5) - 1792*a(n-6) + 1024*a(n-7) - 256*a(n-8).
a(n) - 15*a(n-1) + 98*a(n-2) - 364*a(n-3) + 840*a(n-4) - 1232*a(n-5) + 1120*a(n-6) - 576*a(n-7) + 128*a(n-8) = 27*2^(n-1).

A168306 The fifth left hand column of triangle A167580.

Original entry on oeis.org

105, 6534, 132444, 1593960, 13962848, 98382912, 590814336, 3137815296, 15114950400, 67240622592, 279977837568, 1102376491008, 4137416245248, 14896905748480, 51722619518976, 173913487048704, 568323403481088, 1810359422681088, 5635647921192960

Offset: 5

Johannes W. Meijer, Nov 23 2009

G. C. Greubel, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (20, -180, 960, -3360, 8064, -13440, 15360, -11520, 5120, -1024).

Equals the fifth left hand column of triangle A167580.

Other left hand columns are A014480, A167581, A167582 and A168305.

[2^n*(214*n^9-3963*n^8+30768*n^7-130536*n^6+ 330834*n^5-514332*n^4+484382*n^3-262149*n^2+72342*n- 7560)/241920: n in [5..40]]; // Vincenzo Librandi, Jul 18 2016

LinearRecurrence[{20,-180,960,-3360,8064,-13440,15360,-11520,5120,-1024},{105, 6534, 132444, 1593960, 13962848, 98382912, 590814336, 3137815296, 15114950400, 67240622592},50] (* G. C. Greubel, Jul 17 2016 *)

a(n) = 2^n*(214*n^9 - 3963*n^8 + 30768*n^7 - 130536*n^6 + 330834*n^5 - 514332*n^4 + 484382*n^3 - 262149*n^2 + 72342*n - 7560)/241920.
G.f.: (32*z^5 + 3728*z^4 + 20400*z^3 + 20664*z^2 + 4434*z + 105)/(2*z-1)^10.
a(n) = 20*a(n-1) - 180*a(n-2) + 960*a(n-3) - 3360*a(n-4) + 8064*a(n-5) - 13440*a(n-6) + 15360*a(n-7) - 11520*a(n-8) + 5120*a(n-9) - 1024*a(n-10).
a(n) - 19*a(n-1) + 162*a(n-2) - 816*a(n-3) + 2688*a(n-4) - 6048*a(n-5) + 9408*a(n-6) - 9984*a(n-7) + 6912*a(n-8) - 2816*a(n-9) + 512*a(n-10) = 321*2^(n-1).

A171220 a(n) = (2n + 1)*5^n.

Original entry on oeis.org

1, 15, 125, 875, 5625, 34375, 203125, 1171875, 6640625, 37109375, 205078125, 1123046875, 6103515625, 32958984375, 177001953125, 946044921875, 5035400390625, 26702880859375, 141143798828125, 743865966796875, 3910064697265625, 20503997802734375, 107288360595703125

Offset: 0

Jaume Oliver Lafont, Dec 05 2009

Inserting x=1/sqrt(b) into the power series expansion of arctanh(x) yields the general BBP-type formula log((sqrt(b)+1)/(sqrt(b)-1))*sqrt(b)/2 = Sum_{k>=0} 1/((2k+1)b^k).
This sequence illustrates case b=5, with
Sum_{k>=0} 1/a(k) = sqrt(5)*log((1+sqrt(5))/2).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
David H. Bailey, Compendium of BBP formulas for mathematical constants. See formula 86 at p. 26.
Index entries for linear recurrences with constant coefficients, signature (10,-25).

Cf. A014480 ((2n+1)*2^n), A124647 ((2n+1)*3^n), A058962 ((2n+1)*4^n), A155988 ((2n+1)*9^n), A165283 ((2n+1)*16^n), A166725 ((2n+1)*25^n).

[(2*n+1)*5^n: n in [0..25]]; // Vincenzo Librandi, Jun 08 2011

PARI
```
a(n)=(2*n+1)*5^n
```

a(n) = 10*a(n-1) - 25*a(n-2).
O.g.f: (1+5*x)/(1-5*x)^2.
Sum_{n>=0} (-1)^n/a(n) = sqrt(5)*arctan(1/sqrt(5)). - Amiram Eldar, Feb 26 2022
E.g.f.: exp(5*x)*(1 + 10*x). - Stefano Spezia, May 09 2023

A199299 a(n) = (2n + 1)6^n.

Original entry on oeis.org

1, 18, 180, 1512, 11664, 85536, 606528, 4199040, 28553472, 191476224, 1269789696, 8344332288, 54419558400, 352638738432, 2272560758784, 14575734521856, 93096626946048, 592433080565760, 3757718396731392, 23765029860409344, 149902496042582016, 943288877536247808

Offset: 0

Philippe Deléham, Nov 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-36).

Cf. A014480, A058962, A124647, A155988, A171220, A199300, A199301.

[(2*n+1)*6^n: n in [0..30]]; // Vincenzo Librandi, Nov 05 2011

a[n_] := (2*n + 1)*6^n; Array[a, 25, 0] (* Amiram Eldar, Dec 10 2022 *)

a(n) = (2*n+1)*6^n \\ Amiram Eldar, Dec 10 2022

a(n) = 12*a(n-1) - 36*a(n-2).
G.f.: (1+6*x)/(1-6*x)^2.
a(n) = 6*a(n-1) + 2*6^n. - Vincenzo Librandi, Nov 05 2011
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(6)*arccoth(sqrt(6)).
Sum_{n>=0} (-1)^n/a(n) = sqrt(6)*arccot(sqrt(6)). (End)
E.g.f.: exp(6*x)*(1 + 12*x). - Stefano Spezia, May 07 2023

A073774 Number of plane binary trees of size n+3 and height n.

Original entry on oeis.org

0, 0, 0, 4, 68, 376, 1440, 4736, 14272, 40576, 110592, 291840, 750592, 1890304, 4677632, 11403264, 27443200, 65306624, 153878528, 359399424, 832831488, 1916272640, 4380950528, 9957277696, 22510829568, 50642026496, 113413980160

Offset: 0

Antti Karttunen, Aug 11 2002

nonn

Henry Bottomley and Antti Karttunen, Derivations of the formulas for the diagonals of A073345 & A073346.

Cf. A014480, A073345, A073773, A073775.

A073774 := n -> `if`((n < 3),0,(2^(n-2))*abs(A073775(n-3)));

a(n) = A073345(n+3, n).

a(n < 3) = 0, a(3) = 4, a(n) = 1/12 * 2^(n-1) * (2*n^3 + 9*n^2 - 23*n - 78) or a(n) = 2^(n-2) * |A073775(n-3)| from n >= 3 onward.

A199300 a(n) = (2n + 1)7^n.

Original entry on oeis.org

1, 21, 245, 2401, 21609, 184877, 1529437, 12353145, 98001617, 766718533, 5931980229, 45478515089, 346032180025, 2616003280989, 19668469112621, 147174406808233, 1096686708796833, 8142067989552245, 60251303122686613, 444556912229552577, 3271482918202092041

Offset: 0

Philippe Deléham, Nov 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-49).

Cf. A014480, A058962, A124647, A155988, A171220, A199299, A199301.

[(2*n+1)*7^n: n in [0..30]]; // Vincenzo Librandi, Nov 05 2011

a[n_] := (2*n + 1)*7^n; Array[a, 25, 0] (* Amiram Eldar, Dec 10 2022 *)
LinearRecurrence[{14,-49},{1,21},30] (* Harvey P. Dale, Mar 26 2025 *)

a(n) = (2*n+1)*7^n \\ Amiram Eldar, Dec 10 2022

a(n) = 14*a(n-1) - 49*a(n-2).
G.f.: (1+7*x)/(1-7*x)^2.
a(n) = 7*a(n-1) + 2*7^n. - Vincenzo Librandi, Nov 05 2011
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(7)*arccoth(sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = sqrt(7)*arccot(sqrt(7)). (End)
E.g.f.: exp(7*x)*(1 + 14*x). - Stefano Spezia, May 09 2023

a(15) corrected by Vincenzo Librandi, Nov 05 2011

A199301 a(n) = (2n+1)*8^n.

Original entry on oeis.org

1, 24, 320, 3584, 36864, 360448, 3407872, 31457280, 285212672, 2550136832, 22548578304, 197568495616, 1717986918400, 14843406974976, 127543348822016, 1090715534753792, 9288674231451648, 78812993478983680, 666532744850833408, 5620492334958379008, 47269781688880726016

Offset: 0

Philippe Deléham, Nov 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001018 (Powers of 8), A005408 (2n+1).

Cf. A014480, A058962, A124647, A155988, A171220, A199299, A199300.

[(2*n+1)*8^n: n in [0..30]]; // Vincenzo Librandi, Nov 05 2011

A199301:=n->(2*n+1)*8^n: seq(A199301(n), n=0..20); # Wesley Ivan Hurt, Oct 30 2014

Table[(2 n + 1)*8^n, {n, 0, 20}] (* Wesley Ivan Hurt, Oct 30 2014 *)

a(n) = (2*n+1)*8^n \\ Amiram Eldar, Dec 10 2022

a(n) = 16*a(n-1)-64*a(n-2).
G.f.: (1+8*x)/(1-8*x)^2.
a(n) = 8*(a(n-1)+2^(3*n-2)). - Vincenzo Librandi, Nov 05 2011
a(n) = A005408(n) * A001018(n). - Wesley Ivan Hurt, Oct 30 2014
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(8)*arccoth(sqrt(8)).
Sum_{n>=0} (-1)^n/a(n) = sqrt(8)*arccot(sqrt(8)). (End)
E.g.f.: exp(8*x)*(1 + 16*x). - Stefano Spezia, May 09 2023

a(18) corrected by Vincenzo Librandi, Nov 05 2011

A073773 Number of plane binary trees of size n+2 and height n.

Original entry on oeis.org

0, 0, 0, 6, 40, 152, 480, 1376, 3712, 9600, 24064, 58880, 141312, 333824, 778240, 1794048, 4096000, 9273344, 20840448, 46530560, 103284736, 228065280, 501219328, 1096810496, 2390753280, 5192548352, 11240734720, 24259854336

Offset: 0

Antti Karttunen, Aug 11 2002

nonn

			a(3) = 6 because there exists only these six binary trees of size 5 and height 3:
_\/\/_______\/\/_\/_\/_____\/_\/_\/___\/___V_V___
__\/_\/___\/_\/___\/_\/___\/_\/___\/_\/___\/_\/__
___\./_____\./_____\./_____\./_____\./_____\./___

Henry Bottomley & Antti Karttunen Derivations of the formulas for the diagonals of A073345 & A073346.

Cf. A014480, A073345, A073774, A028878.

A073773 := n -> `if`((n < 3),0,((n^2 - 6)*2^(n-2)));

a(n) = A073345(n+2, n).

a(n < 3) = 0, a(n) = ((n^2 - 6)*2^(n-2)).

cp's OEIS Frontend

A167581 The second left hand column of triangle A167580.

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A167582 The third left hand column of triangle A167580.

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A168305 The fourth left hand column of triangle A167580.

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Magma

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A168306 The fifth left hand column of triangle A167580.

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Magma

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A171220 a(n) = (2n + 1)*5^n.

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Magma

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A199299 a(n) = (2*n + 1)*6^n.

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Magma

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PARI

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A073774 Number of plane binary trees of size n+3 and height n.

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Maple

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A199300 a(n) = (2*n + 1)*7^n.

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Magma

A199299 a(n) = (2n + 1)6^n.

A199300 a(n) = (2n + 1)7^n.