A062196 -id:A062196 - cp's OEIS frontend

A107397 a(n) = binomial(n+6, 6) * binomial(n+8, 6).

28, 588, 5880, 38808, 194040, 792792, 2774772, 8588580, 24048024, 61941880, 148660512, 335785632, 719540640, 1472290848, 2891999880, 5477788008, 10042611348, 17877713700, 30988037080, 52423371000, 86736850200, 140610670200, 223698793500, 349748200620, 538074154800

Offset: 0

Zerinvary Lajos, May 25 2005

			If n=0 then C(0+6,6)*C(0+8,6) = C(6,6)*C(8,6) = 1*28 = 28.
If n=6 then C(6+6,6)*C(6+8,6) = C(12,6)*C(14,6) = 924*3003 = 2774772.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A033987, A062196.

A107397:= func< n | Binomial(n+6,6)*Binomial(n+8,6) >;
[A107397(n): n in [0..30]]; // G. C. Greubel, Feb 09 2025

a[n_] := Binomial[n + 6, 6] * Binomial[n + 8, 6]; Array[a, 25, 0] (* Amiram Eldar, Sep 01 2022 *)

a(n)={binomial(n+6, 6) * binomial(n+8, 6)} \\ Andrew Howroyd, Nov 08 2019

def A107397(n): return binomial(n+6,6)*binomial(n+8,6)
print([A107397(n) for n in range(31)]) # G. C. Greubel, Feb 09 2025

From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 720*Pi^2 - 1740989/245.
Sum_{n>=0} (-1)^n/a(n) = 6144*log(2)/7 - 149046/245. (End)
G.f.: 28*(1 + 8*x + 15*x^2 + 8*x^3 + x^4)/(1-x)^13. - G. C. Greubel, Feb 09 2025

a(3) corrected and terms a(11) and beyond from Andrew Howroyd, Nov 08 2019

A013648 Numbers k such that the periodic part of the continued fraction for sqrt(k) contains a single 1.

Original entry on oeis.org

3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 175, 176, 195, 208, 224, 255, 288, 323, 360, 399, 440, 483, 528, 551, 575, 624, 675, 728, 783, 799, 840, 899, 960, 1023, 1035, 1088, 1155, 1224, 1247, 1295, 1368, 1403, 1443, 1520, 1599, 1680, 1763, 1848, 1872

Offset: 1

Clark Kimberling

nonn

All the terms of A005563 are here, as well as some additional terms (with even period > 2 and the digit 1 in central position) (e.g., sqrt(175) = [13,'4, 2, 1, 2, 4, 26']).

Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Robert Macmillan, Continued fractions, Math. Gaz., Vol. 84, No. 499 (2000), pp. 30-35. See p. 34.

Union of A005563 and A102538.

Cf. A040001, A040005, A040011, A040019, A040029, A062196.

Select[ Range@ 1900, !IntegerQ[ Sqrt@ #] && Count[ ContinuedFraction[ Sqrt@ #][[2]], 1] == 1 &] (* Robert G. Wilson v, Jul 03 2011 *)

Additional comments from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 30 2001

Incorrect formulas and programs removed by R. J. Mathar, Jan 06 2011

A107395 a(n) = binomial(n+4,4)*binomial(n+6,4).

Original entry on oeis.org

15, 175, 1050, 4410, 14700, 41580, 103950, 235950, 495495, 975975, 1821820, 3248700, 5569200, 9224880, 14825700, 23197860, 35441175, 52997175, 77729190, 112015750, 158858700, 222007500, 306101250, 416830050, 561117375, 747325215, 985483800, 1287547800

Offset: 0

Zerinvary Lajos, May 25 2005

			If n=0 then C(0+4,4)*C(0+6,4) = C(4,4)*C(6,4) = 1*15 = 15.
If n=9 then C(9+4,4)*C(9+6,4) = C(13,4)*C(15,4) = 715*1365 = 975975.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A033987, A062196.

A107395:= func< n | Binomial(n+4,4)*Binomial(n+6,4) >;
[A107395(n): n in [0..30]]; // G. C. Greubel, Feb 09 2025

Table[Binomial[n+4,4]Binomial[n+6,4],{n,0,30}] (* Harvey P. Dale, Jun 07 2019 *)

def A107395(n): return binomial(n+4,4)*binomial(n+6,4)
print([A107395(n) for n in range(31)]) # G. C. Greubel, Feb 09 2025

From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 16*Pi^2 - 3946/25.
Sum_{n>=0} (-1)^n/a(n) = 1776/25 - 512*log(2)/5. (End)
G.f.: 5*(3 + 8*x + 3*x^2)/(1-x)^9. - G. C. Greubel, Feb 09 2025

More terms from Harvey P. Dale, Jun 07 2019

A107398 a(n) = binomial(n+7, 7) * binomial(n+9, 7).

Original entry on oeis.org

36, 960, 11880, 95040, 566280, 2718144, 11042460, 39262080, 125147880, 364066560, 979945824, 2466996480, 5859116640, 13220570880, 28506855960, 59025960576, 117846969900, 227667211200, 426876021000, 778861512000, 1386019463400, 2410468632000, 4104188068500

Offset: 0

Zerinvary Lajos, May 25 2005

			If n=0 then C(n+7,7)*C(n+9,7) = C(7,7)*C(9,7) = 1*36 = 36.
If n=4 then C(4+7,7)*C(4+9,7) = C(11,7)*C(13,7) = 330*1716 = 566280.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. A062196.

A107398:= func< n | Binomial(n+7,7)*Binomial(n+9,7) >;
[A107398(n): n in [0..30]]; // G. C. Greubel, Feb 07 2025

a[n_] := Binomial[n + 7, 7] * Binomial[n + 9, 7]; Array[a, 25, 0] (* Amiram Eldar, Sep 01 2022 *)

a(n)={binomial(n+7, 7) * binomial(n+9, 7)} \\ Andrew Howroyd, Nov 08 2019

def A107398(n): return binomial(n+7,7)*binomial(n+9,7)
print([A107398(n) for n in range(31)]) # G. C. Greubel, Feb 07 2025

From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 8085*Pi^2/2 - 12767311/320.
Sum_{n>=0} (-1)^n/a(n) = 245*Pi^2/4 - 580307/960. (End)
G.f.: 12*(3 + 35*x + 105*x^2 + 105*x^3 + 35*x^4 + 3*x^5)/(1-x)^15. - G. C. Greubel, Feb 07 2025

a(3) corrected and terms a(8) and beyond from Andrew Howroyd, Nov 08 2019

A107399 a(n) = binomial(n+8,8)*binomial(n+10,8).

Original entry on oeis.org

45, 1485, 22275, 212355, 1486485, 8281845, 38648610, 156434850, 563165460, 1837398420, 5512195260, 15380181180, 40281426900, 99773995860, 235181561670, 530311364550, 1149007956525, 2401177618125, 4855714738875, 9528883810875, 18191505457125

Offset: 0

Zerinvary Lajos, May 25 2005

			If n=0 then C(0+8,8)*C(0+10,8) = C(8,8)*C(10,8) = 1*45 = 45.
If n=4 then C(7+8,8)*C(7+10,8) = C(15,8)*C(17,8) = 3003*12870 = 38648610.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Cf. A062196.

A107399:= func< n | Binomial(n+8,8)*Binomial(n+10,8) >;
[A107399(n): n in [0..30]]; // G. C. Greubel, Feb 07 2025

Table[Binomial[n+8,8]Binomial[n+10,8],{n,0,20}] (* Harvey P. Dale, Apr 03 2019 *)

def A107399(n): return binomial(n+8,8)*binomial(n+10,8)
print([A107399(n) for n in range(31)]) # G. C. Greubel, Feb 07 2025

From Amiram Eldar, Sep 08 2022: (Start)
Sum_{n>=0} 1/a(n) = 64064*Pi^2/3 - 2987552614/14175.
Sum_{n>=0} (-1)^n/a(n) = 57237184/14175 - 262144*log(2)/45. (End)
G.f.: 45*(1 + 16*x + 70*x^2 + 112*x^3 + 70*x^4 + 16*x^5 + x^6)/(1-x)^17. - G. C. Greubel, Feb 07 2025

More terms from Harvey P. Dale, Apr 03 2019

A105946 a(n) = C(n+3,3) * C(n+5,5).

Original entry on oeis.org

1, 24, 210, 1120, 4410, 14112, 38808, 95040, 212355, 440440, 858858, 1589952, 2815540, 4798080, 7907040, 12651264, 19718181, 30020760, 44753170, 65456160, 94093230, 133138720, 185679000, 255528000, 347358375, 466849656, 620854794, 817586560, 1066825320

Offset: 0

Zerinvary Lajos, Apr 27 2005

			If n=0 then C(0+5,0)*C(0+3,3) = C(5,0)*C(3,3) = 1*1 = 1.
If n=15 then C(15+5,15)*C(15+3,3) = C(20,15)*C(18,3) = 15504*816 = 12651264.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A062196.

A105946:= func< n | Binomial(n+3,3)*Binomial(n+5,5) >;
[A105946(n): n in [0..40]]; // G. C. Greubel, Feb 22 2025

A105946:=n->binomial(n+5,n)*binomial(n+3,3); seq(A105946(n), n=0..100); # Wesley Ivan Hurt, Nov 26 2013

Table[Binomial[n+5,n]*Binomial[n+3,3], {n,0,100}] (* Wesley Ivan Hurt, Nov 26 2013 *)

def A105946(n): return binomial(n+3,3)*binomial(n+5,5)
print([A105946(n) for n in range(41)]) # G. C. Greubel, Feb 22 2025

G.f.: (1 + 15*x + 30*x^2 + 10*x^3)/(1-x)^9. - Colin Barker, Jan 28 2013
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 75*Pi^2/2 - 5905/16.
Sum_{n>=0} (-1)^n/a(n) = 160*log(2) - 5*Pi^2/4 - 4685/48. (End)
E.g.f.: (1/6!)*(720 + 16560*x + 58680*x^2 + 67320*x^3 + 32850*x^4 + 7686*x^5 + 893*x^6 + 49*x^7 + x^8)*exp(x). - G. C. Greubel, Feb 22 2025

More terms from Colin Barker, Jan 28 2013

A105947 a(n) = C(n+4,4) * C(n+6,6).

Original entry on oeis.org

1, 35, 420, 2940, 14700, 58212, 194040, 566280, 1486485, 3578575, 8016008, 16893240, 33786480, 64574160, 118605600, 210327264, 361499985, 604167795, 984569740, 1568220500, 2446423980, 3744526500, 5632263000, 8336601000, 12157543125, 17487410031, 24834191760

Offset: 0

Zerinvary Lajos, Apr 27 2005

			If n=0 then C(0+6,0)*C(0+4,4) = C(6,0)*C(4,4) = 1*1 = 1.
If n=10 then C(10+6,10)*C(10+4,4) = C(16,10)*C(14,4) = 8008*1001 = 8016008.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A062196.

A105947:= func< n | Binomial(n+4,4)*Binomial(n+6,6) >;
[A105947(n): n in [0..40]]; // G. C. Greubel, Feb 22 2025

Table[Binomial[n+6,n]Binomial[n+4,4],{n,0,30}] (* Harvey P. Dale, May 21 2014 *)

def A105947(n): return binomial(n+4,4)*binomial(n+6,6)
print([A105947(n) for n in range(41)]) # G. C. Greubel, Feb 22 2025

G.f.: (1 + 24*x + 90*x^2 + 80*x^3 + 15*x^4)/(1-x)^11. - Colin Barker, Jan 28 2013
From Wesley Ivan Hurt, Jan 27 2022: (Start)
a(n) = (17280 + 78336*n + 152376*n^2 + 167780*n^3 + 116150*n^4 + 52983*n^5 +
   16173*n^6 + 3270*n^7 + 420*n^8 + 31*n^9 + n^10)/17280.
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). (End)
From Amiram Eldar, Sep 08 2022: (Start)
Sum_{n>=0} 1/a(n) = 224*Pi^2 - 55244/25.
Sum_{n>=0} (-1)^n/a(n) = 12*Pi^2 + 512*log(2)/5 - 4711/25. (End)

Terms from a(8) onwards corrected by Colin Barker, Jan 28 2013

Second example corrected by Colin Barker, Jan 28 2013

A105948 a(n) = C(n+5,5) * C(n+7,7).

Original entry on oeis.org

1, 48, 756, 6720, 41580, 199584, 792792, 2718144, 8281845, 22902880, 58402344, 139007232, 311800944, 664191360, 1352103840, 2644114176, 4988699793, 9114302736, 16175074300, 27959131200, 47181033900, 77886151200, 126001769400, 200078424000, 312275179125

Offset: 0

Zerinvary Lajos, Apr 27 2005

			If n=0 then C(0+7,0)*C(0+5,5) = C(7,0)*C(5,5) = 1*1 = 1.
If n=12 then C(12+7,12)*C(12+5,5) = C(19,12)*C(17,5) = 50388*6188 = 311800944.

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A062196.

A105948:= func< n | Binomial(n+5,5)*Binomial(n+7,7) >;
[A105948(n): n in [0..40]]; // G. C. Greubel, Feb 22 2025

Table[Binomial[n+7,n]Binomial[n+5,5],{n,0,30}] (* Harvey P. Dale, Apr 08 2019 *)

def A105948(n): return binomial(n+5,5)*binomial(n+7,7)
print([A105948(n) for n in range(41)]) # G. C. Greubel, Feb 22 2025

G.f.: (1 + 35*x + 210*x^2 + 350*x^3 + 175*x^4 + 21*x^5)/ (1-x)^13. - Colin Barker, Jan 29 2013
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 1225*Pi^2 - 1740851/144.
Sum_{n>=0} (-1)^n/a(n) = 35*Pi^2/6 - 3584*log(2)/3 + 61719/80. (End)

A107319 a(n) = C(n+8,8) * C(n+6,6).

Original entry on oeis.org

1, 63, 1260, 13860, 103950, 594594, 2774772, 11042460, 38648610, 121671550, 350414064, 935402832, 2338507080, 5521090680, 12394285200, 26606398896, 54875697723, 109181751525, 210275965900, 393175282500, 715579014150, 1270517841450, 2205030964500, 3747302149500

Offset: 0

Zerinvary Lajos, May 21 2005

			If n=0 then C(0+8,8)*C(0+6,6) = C(8,8)*C(6,6) = 1*1 = 1.
If n=6 then C(6+8,8)*C(6+6,6) = C(14,8)*C(12,6) = 3003*924 = 2774772.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. A062196.

A107319:= func< n | Binomial(n+6,6)*Binomial(n+8,8) >;
[A107319(n): n in [0..40]]; // G. C. Greubel, Feb 22 2025

Table[Binomial[n+8,8]Binomial[n+6,6],{n,0,20}] (* Harvey P. Dale, Sep 02 2016 *)

def A107319(n): return binomial(n+6,6)*binomial(n+8,8)
print([A107319(n) for n in range(41)]) # G. C. Greubel, Feb 22 2025

From Amiram Eldar, Sep 08 2022: (Start)
Sum_{n>=0} 1/a(n) = 6336*Pi^2 - 76602676/1225.
Sum_{n>=0} (-1)^n/a(n) = 1365906/1225 - 80*Pi^2 - 16384*log(2)/35. (End)

More terms from Harvey P. Dale, Sep 02 2016

A062195 Sixth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).

Original entry on oeis.org

1, 48, 1512, 40320, 997920, 23950080, 570810240, 13699445760, 333923990400, 8310997094400, 211930425907200, 5548723878297600, 149353151057510400, 4135933413900288000, 117874102296158208000

Offset: 0

Wolfdieter Lang, Jun 19 2001

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for sequences related to Laguerre polynomials.

Cf. A001710, A005990, A005461, A062139, A062193, A062194.

Cf. A001113, A001620, A091725, A099285.

[Factorial(n+5)*Binomial(n+7, 7)/Factorial(5): n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+5)!*Binomial[n+7, 7]/5!, {n, 0, 20}] (* G. C. Greubel, May 12 2018 *)

{ f=24; for (n=0, 100, f*=n + 5; write("b062195.txt", n, " ", f*binomial(n + 7, 7)/120) ) } \\ Harry J. Smith, Aug 02 2009

E.g.f.: N(2;5, x)/(1-x)^13 with N(2;5, x) := Sum_{k=0..5} A062196(5, k)*x^k = 1+35*x+210*x^2+350*x^3+175*x^4+21*x^5.
a(n) = A062139(n+5, 5).
a(n) = (n+5)!*binomial(n+7, 7)/5!.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n-5) = (-1)^(n-1)*f(n,5,-8), (n>=5). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 1295*(Ei(1) - gamma) + 2170*e - 22813/3, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=0} (-1)^n/a(n) = 36575*(gamma - Ei(-1)) - 21700/e - 63455/3, where Ei(-1) = -A099285. (End)

cp's OEIS Frontend

A107397 a(n) = binomial(n+6, 6) * binomial(n+8, 6).

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A013648 Numbers k such that the periodic part of the continued fraction for sqrt(k) contains a single 1.

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A107395 a(n) = binomial(n+4,4)*binomial(n+6,4).

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A107398 a(n) = binomial(n+7, 7) * binomial(n+9, 7).

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A107399 a(n) = binomial(n+8,8)*binomial(n+10,8).

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A105946 a(n) = C(n+3,3) * C(n+5,5).

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