A062264 -id:A062264 - cp's OEIS frontend

A104473 a(n) = binomial(n+2,2)*binomial(n+6,2).

15, 63, 168, 360, 675, 1155, 1848, 2808, 4095, 5775, 7920, 10608, 13923, 17955, 22800, 28560, 35343, 43263, 52440, 63000, 75075, 88803, 104328, 121800, 141375, 163215, 187488, 214368, 244035, 276675, 312480, 351648, 394383, 440895, 491400, 546120, 605283, 669123

Offset: 0

Zerinvary Lajos, Apr 18 2005

			a(0) = C(0+2,2)*C(0+6,2) = C(2,2)*C(6,2) = 1*15 = 155.
a(6) = 1*3*5 + 2*4*6 + 3*5*7 + 4*6*8 + 5*7*9 + 6*8*10 + 7*9*11 = 1848.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000579, A033275, A034856, A062264.

Subsequence of A085780.

[Binomial(n+2, 2)*Binomial(n+6, 2): n in [0..50]]; // Vincenzo Librandi, Apr 28 2014

f[n_] := Binomial[n + 2, 2] Binomial[n + 6, 2]; Table[f[n], {n,0,40}] (* Robert G. Wilson v, Apr 20 2005 *)
CoefficientList[Series[3 (5-4*x+x^2)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Apr 28 2014 *)

a(n)=binomial(n+2,2)*binomial(n+6,2) \\ Charles R Greathouse IV, Jun 07 2013

def A104473(n): return binomial(n+2,2)*binomial(n+6,2)
print([A104473(n) for n in range(51)]) # G. C. Greubel, Mar 05 2025

a(n) = (1/4)*(n+1)*(n+2)*(n+5)*(n+6).
a(n) = A034856(n+2)^2 - 1. - J. M. Bergot, Dec 14 2010
G.f.: 3*(5-4*x+x^2)/(1-x)^5. - Colin Barker, Sep 21 2012
a(n) = Sum_{i=1..n+1} i*(i+2)*(i+4). - Bruno Berselli, Apr 28 2014
a(n) = A000217(n)*A000217(n+4) = 3*A033275(n+4). - R. J. Mathar, Nov 29 2015
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 43/450.
Sum_{n>=0} (-1)^n/a(n) = 16*log(2)/15 - 154/225. (End)
From G. C. Greubel, Mar 05 2025: (Start)
a(n) = 90*A000579(n+6)/A000279(n+3).
E.g.f.: (1/4)*(60 + 192*x + 114*x^2 + 20*x^3 + x^4)*exp(x). (End)

A104475 a(n) = binomial(n+4,4) * binomial(n+8,4).

Original entry on oeis.org

70, 630, 3150, 11550, 34650, 90090, 210210, 450450, 900900, 1701700, 3063060, 5290740, 8817900, 14244300, 22383900, 34321980, 51482970, 75710250, 109359250, 155405250, 217567350, 300450150, 409704750, 552210750, 736281000, 971890920, 1270934280, 1647507400, 2118223800

Offset: 0

Zerinvary Lajos, Apr 18 2005

			a(0): C(0+4,4)*C(0+8,4) = C(4,4)*C(8,4) = 1*70 = 70.
a(7): C(5+4,4)*C(5+8,4) = C(9,4)*(13,4) = 126*715 = 90090.

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. A000581, A062264.

[Binomial(n+4,4)*Binomial(n+8,4): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

A104475:=n->binomial(n+4,4)*binomial(n+8,4): seq(A104475(n), n=0..40); # Wesley Ivan Hurt, Jan 29 2017

f[n_] := Binomial[n + 4, 4]Binomial[n + 8, 4]; Table[ f[n], {n, 0, 25}] (* Robert G. Wilson v, Apr 20 2005 *)

vector(30, n, n--; binomial(n+4,4)*binomial(n+8,4)) \\ Michel Marcus, Jul 31 2015

def A104475(n): return binomial(n+4,4)*binomial(n+8,4)
print([A104475(n) for n in range(31)]) # G. C. Greubel, Mar 05 2025

a(n) = 70*A000581(n-8). - Michel Marcus, Jul 31 2015
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 4/245.
Sum_{n>=0} (-1)^n/a(n) = 512*log(2)/35 - 37216/3675. (End)
From G. C. Greubel, Mar 05 2025: (Start)
G.f.: 70/(1-x)^9.
E.g.f.: (1/576)*(40320 + 322560*x + 564480*x^2 + 376320*x^3 + 117600*x^4 + 18816*x^5 + 1568*x^6 + 64*x^7 + x^8)*exp(x). (End)

More terms from Robert G. Wilson v, Apr 20 2005

A104476 a(n) = binomial(n+7,7)*binomial(n+11,7).

Original entry on oeis.org

330, 6336, 61776, 411840, 2123550, 9060480, 33372768, 109219968, 324246780, 886828800, 2261413440, 5427392256, 12352970916, 26829982080, 55895796000, 112183843200, 217706770710, 409800980160, 750266946000, 1339149240000, 2335141487250, 3985308138240

Offset: 0

Zerinvary Lajos, Apr 18 2005

			a(0): C(0+7,7)*C(0+11,7) = C(7,7)*C(11,7) = 1*330 = 330;
a(7): C(7+7,7)*C(7+11,7) = C(14,7)*C(18,7) = 3432*31824 = 109219968.

T. D. Noe, 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. A062264.

[Binomial(n+7,7)*Binomial(n+11,7): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

f[n_] := Binomial[n + 7, 7]*Binomial[n + 11, 7]; Table[ f[n], {n, 0, 19}] (* Robert G. Wilson v, Apr 20 2005 *)

vector(30, n, n--; binomial(n+7,7)*binomial(n+11,7)) \\ Michel Marcus, Jul 31 2015

A104476_list, m = [], [3432, -1716, 660, 330, 330, 330, 330, 330, 330, 330, 330, 330, 330, 330, 330]
for _ in range(10**2):
    A104476_list.append(m[-1])
    for i in range(14):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

def A104476(n): return binomial(n+7,7)*binomial(n+11,7)
print([A104476(n) for n in range(31)]) # G. C. Greubel, Mar 04 2025

From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 539*Pi^2 - 114905813/21600.
Sum_{n>=0} (-1)^n/a(n) = 1741019/7200 - 49*Pi^2/2. (End)
G.f.: 66*(5 + 21*x + 21*x^2 + 5*x^3)/(1-x)^15. - G. C. Greubel, Mar 04 2025

More terms from Robert G. Wilson v, Apr 20 2005

A104478 a(n) = binomial(n+8,8)*binomial(n+12,8).

Original entry on oeis.org

495, 11583, 135135, 1061775, 6370650, 31286970, 131405274, 486370170, 1621233900, 4946841900, 13992495660, 37058912748, 92647281870, 220089696750, 499568676750, 1088533853550, 2285921092455, 4642276728375, 9143878404375, 17513561154375, 32691980821500, 59592810754620

Offset: 0

Zerinvary Lajos, Apr 18 2005

All terms are multiples of 99. - Michel Marcus, Aug 01 2015

			a(0): C(0+8,8)*C(0+12,8) = C(8,8)*C(12,8) = 1*495 = 495.
a(7): C(7+8,8)*C(7+12,8) = C(15,8)*C(19,8) = 6435*75582 = 486370170.

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. A000581, A062264.

[Binomial(n+8,8)*Binomial(n+12,8): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

f[n_] := Binomial[n + 8, 8] * Binomial[n + 12, 8]; Table[ f[n], {n, 0, 18}] (* Robert G. Wilson v, Apr 19 2005 *)

vector(30, n, n--; binomial(n+8,8)*binomial(n+12,8)) \\ Michel Marcus, Jul 31 2015

def A104478(n): return binomial(n+8,8)*binomial(n+12,8)
print([A104478(n) for n in range(31)]) # G. C. Greubel, Mar 04 2025

a(n) = A000581(n+8)*A000581(n+12). - Michel Marcus, Aug 01 2015
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=0} 1/a(n) = 11648*Pi^2/3 - 65726161036/1715175.
Sum_{n>=0} (-1)^n/a(n) = 262144*log(2)/99 - 629604992/343035. (End)
G.f.: 99*(5 + 32*x + 56*x^2 + 32*x^3 + 5*x^4)/(1-x)^17. - G. C. Greubel, Mar 04 2025

Corrected and extended by Robert G. Wilson v, Apr 19 2005

a(6) corrected by Georg Fischer, May 08 2021

A105249 a(n) = binomial(n+2,n)*binomial(n+6,n).

Original entry on oeis.org

1, 21, 168, 840, 3150, 9702, 25872, 61776, 135135, 275275, 528528, 965328, 1689324, 2848860, 4651200, 7379904, 11415789, 17261937, 25573240, 37191000, 53183130, 74890530, 103980240, 142506000, 192976875, 258434631, 342540576, 449672608, 585033240, 754769400

Offset: 0

Zerinvary Lajos, Apr 14 2005

			a(0): C(0+2,0)*C(0+6,0) = C(2,0)*C(6,0) = 1*1 = 1;
a(10): C(10+2,10)*C(10+6,10) = C(12,10)*C(16,10) = 66*8008 = 528528.

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. A062264.

[Binomial(n+2,n)*Binomial(n+6,n): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

f[n_] := Binomial[n + 2, n]Binomial[n + 6, n]; Table[ f[n], {n, 0, 27}] (* Robert G. Wilson v, Apr 20 2005 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,21,168,840,3150,9702,25872,61776,135135},30] (* Harvey P. Dale, Oct 08 2012 *)

def A105249(n): return binomial(n+2,n)*binomial(n+6,n)
print([a105249(n) for n in range(31)]) # G. C. Greubel, Mar 04 2025

a(0)=1, a(1)=21, a(2)=168, a(3)=840, a(4)=3150, a(5)=9702, a(6)=25872, a(7)=61776, a(8)=135135, a(n) = 9*a(n-1) -36*a(n-2) +84*a(n-3) -126*a(n-4) +126*a(n-5) -84*a(n-6) +36*a(n-7) -9*a(n-8) +a(n-9). - Harvey P. Dale, Oct 08 2012
G.f.: (1+12*x+15*x^2)/(1-x)^9. - Colin Barker, Jan 21 2013
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=0} 1/a(n) = 12*Pi^2 - 5869/50.
Sum_{n>=0} (-1)^n/a(n) = 256*log(2)/5 - 4*Pi^2 + 371/75. (End)
E.g.f.: (1/1440)*(1440 + 28800*x + 91440*x^2 + 95520*x^3 + 42900*x^4 + 9312*x^5 + 1010*x^6 + 52*x^7 + x^8)*exp(x). - G. C. Greubel, Mar 04 2025

More terms from Robert G. Wilson v, Apr 20 2005

A105250 a(n) = binomial(n+3,n)*binomial(n+7,n).

Original entry on oeis.org

1, 32, 360, 2400, 11550, 44352, 144144, 411840, 1061775, 2516800, 5562128, 11583936, 22926540, 43411200, 79070400, 139163904, 237557133, 394558560, 639331000, 1013012000, 1572701130, 2396496960, 3589794000, 5293080000, 7691506875, 11026544256, 15610063392

Offset: 0

Zerinvary Lajos, Apr 14 2005

			a(0): C(0+3,0)*C(0+7,0) = C(3,0)*C(7,0) = 1*1 = 1;
a(10): C(10+3,10)*C(10+7,10) = C(13,10)*(17,10) = 286*19448 = 5562128.

T. D. Noe, 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. A062264.

[Binomial(n+3,n)*Binomial(n+7,n): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

f[n_] := Binomial[n + 3, n]Binomial[n + 7, n]; Table[ f[n], {n, 0, 23}] (* Robert G. Wilson v, Apr 20 2005 *)

def A105250(n): return binomial(n+3,n)*binomial(n+7,n)
print([A105250(n) for n in range(31)]) # G. C. Greubel, Mar 04 2025

G.f.: (1+21*x+63*x^2+35*x^3)/(1-x)^11. - Colin Barker, Jan 21 2013
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). - Wesley Ivan Hurt, May 24 2021
From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 98*Pi^2 - 72464/75.
Sum_{n>=0} (-1)^n/a(n) = 7*Pi^2 + 1792*log(2)/5 - 15827/50. (End)

More terms from Robert G. Wilson v, Apr 20 2005

A105252 a(n) = binomial(n+5,n)*binomial(n+9,n).

Original entry on oeis.org

1, 60, 1155, 12320, 90090, 504504, 2312310, 9060480, 31286970, 97337240, 277411134, 733649280, 1818838840, 4261894560, 9502285320, 20271542016, 41572498275, 82281899700, 157706974425, 293570877600, 532097215650, 941124327000, 1627522854750, 2756636064000

Offset: 0

Zerinvary Lajos, Apr 14 2005

			a(0): C(0+5,0)*C(0+9,0) = C(5,0)*C(9,0) = 1*1 = 1;
a(10): C(10+5,10)*C(10+9,10) = C(15,10)*(19,10) = 3003*92378 = 277411134.

T. D. Noe, 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. A062264.

[Binomial(n+5,n)*Binomial(n+9,n): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

f[n_] := Binomial[n + 5, n]Binomial[n + 9, n]; Table[ f[n], {n, 0, 20}] (* Robert G. Wilson v, Apr 20 2005 *)

A105252_list, m = [], [2002, -4433, 3487, -1133, 127, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
for _ in range(10**2):
    A105252_list.append(m[-1])
    for i in range(14):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

def A105252(n): return binomial(n+5,n)*binomial(n+9,n)
print([A105252(n) for n in range(31)]) # G. C. Greubel, Mar 04 2025

G.f.: (1+45*x+360*x^2+840*x^3+630*x^4+126*x^5)/(1-x)^15. - Colin Barker, Jan 21 2013
From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 7425*Pi^2/2 - 114902691/3136.
Sum_{n>=0} (-1)^n/a(n) = 28960047/15680 - 255*Pi^2/4 - 12288*log(2)/7. (End)

More terms from Robert G. Wilson v, Apr 20 2005

A105253 a(n) = binomial(n+6,n)*binomial(n+10,n).

Original entry on oeis.org

1, 77, 1848, 24024, 210210, 1387386, 7399392, 33372768, 131405274, 462351890, 1479526048, 4365213216, 12004336344, 31040798712, 76018282560, 177375992640, 396324483555, 851617661895, 1766318113560, 3547314771000, 6917263803450, 13128684361650, 24304341297600

Offset: 0

Zerinvary Lajos, Apr 14 2005

			a(0): C(0+6,0)*C(0+10,0) = C(6,0)*C(10,0) = 1*1 = 1;
a(10): C(10+6,10)*C(10+10,10) = C(16,10)*(20,10) = 8008*184756 = 1479526048.

T. D. Noe, 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. A062264.

[Binomial(n+6,n)*Binomial(n+10,n): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

f[n_] := Binomial[n + 6, n]Binomial[n + 10, n]; Table[ f[n], {n, 0, 20}] (* Robert G. Wilson v, Apr 20 2005 *)

A105253_list, m = [], [8008, -22022, 23023, -11297, 2563, -209] + [1]*11
for _ in range(10**2):
    A105253_list.append(m[-1])
    for i in range(16):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

def A105253(n): return binomial(n+6,n)*binomial(n+10,n)
print([A105253(n) for n in range(31)]) # G. C. Greubel, Mar 04 2025

G.f.: (1 + 60*x + 675*x^2 + 2400*x^3 + 3150*x^4 + 1513*x^5 + 210*x^6)/(1-x)^17. - Colin Barker, Jan 21 2013
From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 20020*Pi^2 - 1493768807/7560.
Sum_{n>=0} (-1)^n/a(n) = 131072*log(2)/21 - 100*Pi^2 - 88332653/26460. (End)

More terms from Robert G. Wilson v, Apr 20 2005

A105254 a(n) = binomial(n+7,n)*binomial(n+11,n).

Original entry on oeis.org

1, 96, 2808, 43680, 450450, 3459456, 21237216, 109219968, 486370170, 1921462400, 6859620768, 22449667968, 68128506264, 193501082880, 518306472000, 1317650231040, 3196331224515, 7432299594720, 16630917303000, 35933837940000, 75191555889450, 152770145299200

Offset: 0

Zerinvary Lajos, Apr 14 2005

			a(0): C(0+7,0)*C(0+11,0) = C(7,0)*C(11,0) = 1*1 = 1;
a(8): C(8+7,8)*C(8+11,8) = C(15,8)*(19,8) = 6435*75582 = 486370170.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (19,-171,969,-3876,11628,-27132,50388,-75582,92378,-92378,75582,-50388,27132,-11628,3876,-969,171,-19,1).

Cf. A062264.

[Binomial(n+7,n)*Binomial(n+11,n): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

f[n_] := Binomial[n + 7, n]Binomial[n + 11, n]; Table[ f[n], {n, 0, 19}] (* Robert G. Wilson v, Apr 20 2005 *)

def A105254(n): return binomial(n+7,n)*binomial(n+11,n)
print([A105254(n) for n in range(31)]) # G. C. Greubel, Mar 04 2025

G.f.: (1 + 77*x + 1155*x^2 + 5775*x^3 + 11550*x^4 + 9702*x^5 + 3234*x^6 + 330*x^7)/(1-x)^19. - Colin Barker, Jan 21 2013
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=0} 1/a(n) = 308308*Pi^2/3 - 16431524791/16200.
Sum_{n>=0} (-1)^n/a(n) = 1232*Pi^2/3 + 360448*log(2)/45 - 108911693/11340. (End)

More terms from Robert G. Wilson v, Apr 20 2005

More terms from Colin Barker, Jan 21 2013

cp's OEIS Frontend

A104473 a(n) = binomial(n+2,2)*binomial(n+6,2).

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A104475 a(n) = binomial(n+4,4) * binomial(n+8,4).

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A104476 a(n) = binomial(n+7,7)*binomial(n+11,7).

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A104478 a(n) = binomial(n+8,8)*binomial(n+12,8).

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A105249 a(n) = binomial(n+2,n)*binomial(n+6,n).

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Magma

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A105250 a(n) = binomial(n+3,n)*binomial(n+7,n).

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