cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

A137954 G.f. satisfies A(x) = 1 + x + x^2*A(x)^4.

Original entry on oeis.org

1, 1, 1, 4, 10, 32, 107, 360, 1270, 4544, 16537, 61092, 228084, 860056, 3269994, 12521488, 48250690, 186959312, 727989318, 2847167632, 11179394088, 44053232012, 174160578150, 690576010820, 2745713062854, 10944253432600
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137955, A137953; A019497, A137959, A137966.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n-k,k]/(n-k)*Binomial[4*k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Sep 18 2013 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x+x^2*A^4);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-k,k)/(n-k)*binomial(4*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

a(n) = Sum_{k=0..n-1} C(n-k,k)/(n-k) * C(4*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
Recurrence: 3*(n-1)*n*(3*n-8)*(3*n-5)*(3*n-2)*(3*n+2)*a(n) = 64*(n-1)^2*(2*n-3)*(2*n-1)*(3*n-8)*(3*n-5)*a(n-2) + 32*(2*n-3)*(3*n-8)*(36*n^4 - 204*n^3 + 364*n^2 - 216*n + 35)*a(n-3) + 16*(3*n-2)*(144*n^5 - 1536*n^4 + 6005*n^3 - 10278*n^2 + 6790*n - 600)*a(n-4) + 8*n*(2*n-7)*(3*n-5)*(3*n-2)*(4*n-19)*(4*n-9)*a(n-5). - Vaclav Kotesovec, Sep 18 2013
a(n) ~ sqrt(s*(1-s)*(4-5*s) / ((24*s - 24)*Pi)) / (n^(3/2) * r^n), where r = 0.2362629484147719796376166796890824064312524895955... and s = 1.648350597886362639516822239585443208575003319460... are real roots of the system of equations s = 1 + r*(1 + r*s^4), 4 * r^2 * s^3 = 1. - Vaclav Kotesovec, Nov 22 2017

A137966 G.f. satisfies A(x) = 1+x + x^2*A(x)^6.

Original entry on oeis.org

1, 1, 1, 6, 21, 86, 396, 1812, 8607, 41958, 207333, 1040234, 5281965, 27078756, 140021248, 729369474, 3823598232, 20158251814, 106809280563, 568471343322, 3037782047947, 16292380484454, 87669285293451, 473172657154822
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137967, A137965; A019497, A137954, A137959.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[Binomial[n-k,k]/(n-k) * Binomial[6*k,n-k-1], {k,0,n-1}], {n,1,30}]}] (* Vaclav Kotesovec, Nov 18 2017 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^6)^1);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-k,k)/(n-k)*binomial(6*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

a(n) = Sum_{k=0..n-1} C(n-k,k)/(n-k) * C(6*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
Recurrence: 5*(n-1)*n*(5*n - 26)*(5*n - 21)*(5*n - 19)*(5*n - 16)*(5*n - 14)*(5*n - 12)*(5*n - 11)*(5*n - 9)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 2)*(5*n + 2)*a(n) = + 576*(n-1)^2*(3*n - 5)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(5*n - 26)*(5*n - 21)*(5*n - 19)*(5*n - 16)*(5*n - 14)*(5*n - 12)*(5*n - 11)*(5*n - 9)*(5*n - 7)*a(n-2) + 288*(3*n - 5)*(3*n - 4)*(5*n - 26)*(5*n - 21)*(5*n - 19)*(5*n - 16)*(5*n - 14)*(5*n - 12)*(11250*n^7 - 111375*n^6 + 440175*n^5 - 888545*n^4 + 975241*n^3 - 574177*n^2 + 165869*n - 18018)*a(n-3) + 144*(5*n - 26)*(5*n - 21)*(5*n - 19)*(5*n - 6)*(10125000*n^11 - 218700000*n^10 + 2075585625*n^9 - 11378954250*n^8 + 39836289925*n^7 - 92894908470*n^6 + 145953551806*n^5 - 152681445300*n^4 + 102505633480*n^3 - 41086190160*n^2 + 8557182144*n - 670602240)*a(n-4) + 72*(5*n - 26)*(5*n - 11)*(5*n - 6)*(5*n - 4)*(20250000*n^11 - 569025000*n^10 + 7025658750*n^9 - 50083579125*n^8 + 227686012400*n^7 - 687547140050*n^6 + 1391232445598*n^5 - 1854143517725*n^4 + 1550931293540*n^3 - 737424345140*n^2 + 162058858752*n - 10360465920)*a(n-5) + 144*(5*n - 16)*(5*n - 11)*(5*n - 9)*(5*n - 6)*(5*n - 4)*(5*n - 2)*(202500*n^9 - 5953500*n^8 + 74924775*n^7 - 526434885*n^6 + 2255339082*n^5 - 6025054075*n^4 + 9796892735*n^3 - 8893818500*n^2 + 3545754268*n - 142331280)*a(n-6) + 72*n*(2*n - 9)*(3*n - 17)*(3*n - 10)*(5*n - 21)*(5*n - 16)*(5*n - 14)*(5*n - 11)*(5*n - 9)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 2)*(6*n - 41)*(6*n - 13)*a(n-7). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt((1 + 2*r*s^6)/(15*Pi)) / (2*s^2 * n^(3/2) * r^(n + 1/2)), where r = 0.1734895129039028676461340698295316044509963479582... and s = 1.408187415484683441175360883795437925341195617549... are roots of the system of equations 1 + r + r^2*s^6 = s, 6*r^2*s^5 = 1. - Vaclav Kotesovec, Nov 18 2017

A182454 G.f. satisfies A(x) = 1 + x*A(x) + x^2*A(x)^5.

Original entry on oeis.org

1, 1, 2, 7, 27, 112, 492, 2243, 10513, 50353, 245353, 1212398, 6061225, 30601910, 155808915, 799096655, 4124491215, 21408066097, 111672838857, 585128521777, 3078178384457, 16252057372887, 86089680204939, 457400940705274, 2436895852070559, 13015917111573039
Offset: 0

Views

Author

Paul D. Hanna, Apr 29 2012

Keywords

Comments

Compare to a g.f. C(x) of Catalan numbers: C(x) = 1 + x*C(x) + x^2*C(x)^3.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 27*x^4 + 112*x^5 + 492*x^6 +..
Related expansions:
A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 141*x^4 + 615*x^5 + 2792*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 241*x^4 + 1080*x^5 + 4998*x^6 +...
A(x)^5 = 1 + 5*x + 20*x^2 + 85*x^3 + 380*x^4 + 1751*x^5 + 8270*x^6 +...

Links

Crossrefs

Cf. A019497.
Cf. A000045, A000108, A001006, A186996.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*A+x^2*A^5+x*O(x^n));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A)*(1+x*A^4)/((1+x*A^3)*1+x*O(x^n))); polcoeff(A, n)}
  • PARI
    {a(n)=polcoeff((1/x)*serreverse(x^2/serreverse((sqrt(1+4*x-4*x^3+x^2*O(x^n))-1)/2)),n)}
  • PARI
    {a(n)=polcoeff(sqrt((1/x)*serreverse((1+2*x-2*x^3-sqrt(1+4*x-4*x^3+x^3*O(x^n)))/(2*x))),n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f.: A(x) = sqrt( (1/x)*Series_Reversion( (1 + 2*x - 2*x^3 - sqrt(1 + 4*x - 4*x^3))/(2*x) ) ).
G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A019497 (number of ternary search trees on n keys).
G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x*A(x)^4) / (1 + x*A(x)^3).
Recurrence: 64*(n-1)*n*(2*n - 1)*(2*n + 1)*(5*n - 17)*(5*n - 14)*(5*n - 12)*(5*n - 11)*(5*n - 9)*(5*n - 7)*a(n) = 32*(n-1)*(2*n - 1)*(5*n - 17)*(5*n - 14)*(5*n - 12)*(2500*n^5 - 16000*n^4 + 37400*n^3 - 38660*n^2 + 16767*n - 2304)*a(n-1) + (5*n - 17)*(5*n - 8)*(353125*n^8 - 4590625*n^7 + 26079625*n^6 - 84463075*n^5 + 169363570*n^4 - 212446228*n^3 + 159705192*n^2 - 64147968*n + 10184832)*a(n-2) + 8*(5*n - 2)*(1000000*n^9 - 19100000*n^8 + 158791250*n^7 - 752940875*n^6 + 2239835525*n^5 - 4325771435*n^4 + 5410989493*n^3 - 4216402206*n^2 + 1852118136*n - 348425280)*a(n-3) - 8*(5*n - 7)*(5*n - 4)*(5*n - 2)*(20000*n^7 - 368000*n^6 + 2847450*n^5 - 11988080*n^4 + 29592479*n^3 - 42711795*n^2 + 33256206*n - 10724400)*a(n-4) + 8*(n-4)*(2*n - 5)*(4*n - 19)*(4*n - 13)*(5*n - 12)*(5*n - 9)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 2)*a(n-5). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt((1 + 2*r*s^4) / (10*Pi)) / (2*s * n^(3/2) * r^(n + 1/2)), where r = 0.1762643878022406506907195466376048222228890731329... and s = 1.517477187449684643254531724911215527841313263152... are roots of the system of equations 1 + r*s + r^2*s^5 = s, r + 5*r^2*s^4 = 1. - Vaclav Kotesovec, Nov 18 2017
a(n) = Sum_{k=0..floor(n/2)} binomial(n+3*k,k) * binomial(n+2*k,n-2*k) / (4*k+1). - Seiichi Manyama, Jul 26 2023

A137959 G.f. satisfies A(x) = 1 + x + x^2*A(x)^5.

Original entry on oeis.org

1, 1, 1, 5, 15, 55, 220, 876, 3645, 15485, 66735, 292155, 1293456, 5782320, 26071435, 118402495, 541150155, 2487204315, 11488482130, 53302256250, 248293549685, 1160794446445, 5444674773325, 25614768620105, 120837493137460
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137960, A137958; A019497, A137954, A137966.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n-k,k]/(n-k)*Binomial[5*k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Sep 18 2013 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5));polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-k,k)/(n-k)*binomial(5*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

a(n) = Sum_{k=0..n-1} C(n-k,k)/(n-k) * C(5*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
Recurrence: 64*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n-5)*(2*n-3)*(2*n-1)*(2*n+1)*a(n) = + 5*(n-4)*(n-3)*(n-2)*(2*n-5)*(2*n-3)*(5*n-8)*(5*n-6)*(5*n-4)*(5*n-2)*a(n-2) + 5*(n-4)*(n-3)*(2*n-5)*(5000*n^6 - 45000*n^5 + 157250*n^4 - 267750*n^3 + 227216*n^2 - 87057*n + 11520)*a(n-3) + 15*(n-4)*(5000*n^8 - 80000*n^7 + 532250*n^6 - 1903250*n^5 + 3938648*n^4 - 4710638*n^3 + 3044313*n^2 - 895443*n + 80640)*a(n-4) + 5*(n-2)*(2*n-1)*(5000*n^7 - 95000*n^6 + 734250*n^5 - 2951750*n^4 + 6510194*n^3 - 7505289*n^2 + 3655107*n - 207360)*a(n-5) + 5*(n-3)*(n-2)*n*(2*n-3)*(2*n-1)*(5*n-29)*(5*n-23)*(5*n-17)*(5*n-11)*a(n-6). - Vaclav Kotesovec, Sep 18 2013
a(n) ~ sqrt(s*(1-s)*(5-6*s) / ((40*s - 40)*Pi)) / (n^(3/2) * r^n), where r = 0.1990700277700792324868112833575428736312653553870... and s = 1.498837534712599040608514104196928592039081694233... are real roots of the system of equations s = 1 + r*(1 + r*s^5), 5 * r^2 * s^4 = 1. - Vaclav Kotesovec, Nov 22 2017

A137952 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^3)^2.

Original entry on oeis.org

1, 1, 2, 7, 24, 95, 386, 1641, 7150, 31844, 144216, 662228, 3076044, 14427582, 68235078, 325049475, 1558212804, 7511319253, 36387218312, 177050945886, 864912345340, 4240388439744, 20857232340528, 102896737106415
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Crossrefs

Cf. A137953, A019497; A137955, A137960, A137967.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[Binomial[2*(n-k),k]/(n-k) * Binomial[3*k,n-k-1], {k,0,n-1}], {n,1,30}]}] (* Vaclav Kotesovec, Nov 18 2017 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^3)^2);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(2*(n-k),k)/(n-k)*binomial(3*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^2 where B(x) is the g.f. of A137953.
a(n) = Sum_{k=0..n-1} C(2*(n-k),k)/(n-k) * C(3*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
Recurrence: 5*n*(5*n - 4)*(5*n - 3)*(5*n - 1)*(5*n + 3)*(8845200*n^11 - 252428400*n^10 + 3221192232*n^9 - 24137808840*n^8 + 117463352781*n^7 - 387964460127*n^6 + 882822962553*n^5 - 1374856808005*n^4 + 1422227015434*n^3 - 915895407668*n^2 + 320324023880*n - 42693386400)*a(n) = - 360*(5*n - 2)*(5670000*n^13 - 63714600*n^12 - 1032645960*n^11 + 24848001198*n^10 - 218480624507*n^9 + 1101741928166*n^8 - 3582401014336*n^7 + 7865579681092*n^6 - 11836392808433*n^5 + 12130520012664*n^4 - 8236278842764*n^3 + 3497924862840*n^2 - 827741189520*n + 81691545600)*a(n-1) + 180*(2653560000*n^16 - 86342760000*n^15 + 1284348733200*n^14 - 11544882534000*n^13 + 69915022739748*n^12 - 301277354913324*n^11 + 951521048997123*n^10 - 2235356609743737*n^9 + 3921814538564296*n^8 - 5108337175422974*n^7 + 4854490688899951*n^6 - 3250616687965913*n^5 + 1431302003002666*n^4 - 349408874612852*n^3 + 16089460853736*n^2 + 12240998632800*n - 2031289747200)*a(n-2) + 72*(17672709600*n^16 - 601551846000*n^15 + 9383367519936*n^14 - 88661500185240*n^13 + 565349613141438*n^12 - 2565633937621131*n^11 + 8513410651166583*n^10 - 20875837005697545*n^9 + 37705724089968084*n^8 - 49181218885648923*n^7 + 44098626888119141*n^6 - 23771481353637565*n^5 + 3467317211974378*n^4 + 4824415011450004*n^3 - 3654086377331160*n^2 + 1070168332564800*n - 116760296016000)*a(n-3) + 144*(8597534400*n^16 - 305543145600*n^15 + 4975684360704*n^14 - 49077873815616*n^13 + 326509076764188*n^12 - 1543742190898488*n^11 + 5321067950386782*n^10 - 13479709842928188*n^9 + 24903384308348709*n^8 - 32579354322085314*n^7 + 27941366702438094*n^6 - 11913061039189846*n^5 - 3157851308946897*n^4 + 7647346836930652*n^3 - 4534021704525180*n^2 + 1245319349576400*n - 132684717816000)*a(n-4) + 72*(2*n - 7)*(3*n - 14)*(3*n - 10)*(6*n - 25)*(6*n - 23)*(8845200*n^11 - 155131200*n^10 + 1183394232*n^9 - 5046898752*n^8 + 12951310413*n^7 - 19922972292*n^6 + 16394061984*n^5 - 2858995378*n^4 - 7011543813*n^3 + 6369403462*n^2 - 2180183136*n + 267092640)*a(n-5). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt((1 + 4*r*s^3 + 3*r^2*s^6) / (3*Pi*s*(2 + 5*r*s^3))) / (2*n^(3/2) * r^(n + 1/2)), where r = 0.1898739884773465982357897900946346962414966313829... and s = 1.607584028097173055359903977736399386285943742600... are roots of the system of equations 1 + r*(1 + r*s^3)^2 = s, 6*r^2*s^2*(1 + r*s^3) = 1. - Vaclav Kotesovec, Nov 18 2017

A365128 G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x)))^3.

Original entry on oeis.org

1, 3, 15, 88, 567, 3876, 27607, 202653, 1522365, 11647038, 90435804, 710855544, 5645365576, 45228648078, 365109237801, 2966862631856, 24248879149005, 199213507774365, 1644138419038500, 13625326165675698, 113336685917785332, 945931091151789808
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2023

Keywords

Crossrefs

Cf. A001006, A143927.
Cf. A019497, A255673.

Programs

  • Maple
    A365128 := proc(n)
    add(binomial(3*(n+1),k) * binomial(k,n-k),k=0..n) ;
    %/(n+1) ;
end proc:
seq(A365128(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    a(n, s=1, t=3) = sum(k=0, n, binomial(t*(n+1), k)*binomial(s*k, n-k))/(n+1);

Formula

If g.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^s)^t, then a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(t*(n+1),k) * binomial(s*k,n-k).
D-finite with recurrence 205*(5*n+6)*(5*n+2)*(5*n+3)*(5*n+4)*(n+1)*a(n) +9*(-5948592*n^5+11369145*n^4 -5182620*n^3 -351495*n^2+204302*n-6560) *a(n-1) +243*(-801282*n^5 +14391105*n^4 -55889790*n^3 +90254895*n^2 -66199848*n +18182560)*a(n-2) +6561*(3*n-5) *(3*n-4)*(93048*n^3 -579621*n^2 +1227037*n -878874)*a(n-3) +48715425*(n-3) *(3*n-4)*(3*n-7) *(3*n-5)*(3*n-8)*a(n-4)=0. - R. J. Mathar, Dec 04 2023
From Seiichi Manyama, Sep 20 2024: (Start)
G.f.: (1/x) * Series_Reversion( x / (1+x+x^2)^3 ).
G.f.: B(x)^3, where B(x) is the g.f. of A255673. (End)

A366555 G.f. A(x) satisfies A(x) = 1 + x + x^3*A(x)^3.

Original entry on oeis.org

1, 1, 0, 1, 3, 3, 4, 15, 30, 42, 99, 255, 475, 915, 2232, 4977, 9945, 21945, 51093, 110634, 238005, 542341, 1227390, 2696841, 6035886, 13770402, 31001133, 69485295, 157945293, 359888373, 814699002, 1850816823, 4231092060, 9659302380, 22028018679
Offset: 0

Views

Author

Seiichi Manyama, Oct 13 2023

Keywords

Crossrefs

Cf. A019497, A366266, A366556.
Cf. A366591.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(2*k+1, n-3*k)*binomial(3*k, k)/(2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*k+1,n-3*k) * binomial(3*k,k)/(2*k+1).
a(n) = A366591(n) + A366591(n-1).

A366556 G.f. A(x) satisfies A(x) = 1 + x + x^4*A(x)^3.

Original entry on oeis.org

1, 1, 0, 0, 1, 3, 3, 1, 3, 15, 30, 30, 27, 87, 252, 420, 475, 747, 2064, 4632, 7203, 9933, 19635, 47025, 92013, 144745, 237510, 498498, 1073817, 1969131, 3267411, 5977881, 12462579, 25035747, 45090936, 79414344, 153115299, 311198457, 600883569, 1090988379, 2012793705
Offset: 0

Views

Author

Seiichi Manyama, Oct 13 2023

Keywords

Crossrefs

Cf. A019497, A366266, A366555.
Cf. A366554, A366558.
Cf. A366592.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(2*k+1, n-4*k)*binomial(3*k, k)/(2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(2*k+1,n-4*k) * binomial(3*k,k)/(2*k+1).
a(n) = A366592(n) + A366592(n-1).

A346733 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x)^3.

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 10, 21, 48, 103, 219, 489, 1114, 2517, 5712, 13152, 30492, 70812, 165165, 387456, 912378, 2154250, 5102343, 12123027, 28878384, 68947041, 164979006, 395604531, 950428335, 2287387152, 5514240673, 13314167718, 32194109193, 77953239507, 188997294360
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 30 2021

Keywords

Crossrefs

Cf. A001764, A019497, A307970, A346734, A346735.

Programs

  • Mathematica
    nmax = 34; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 3, 1, Sum[Sum[a[i] a[j] a[n - i - j - 3], {j, 0, n - i - 3}], {i, 0, n - 3}]]; Table[a[n], {n, 0, 34}]

Formula

a(0) = a(1) = a(2) = 1; a(n) = Sum_{i=0..n-3} Sum_{j=0..n-i-3} a(i) * a(j) * a(n-i-j-3).

A346734 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x)^3.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 6, 10, 15, 27, 55, 111, 210, 388, 741, 1473, 2956, 5856, 11514, 22806, 45756, 92394, 186459, 375867, 759519, 1541803, 3140775, 6407307, 13081230, 26745378, 54797850, 112495734, 231270690, 475960278, 980643070, 2023057266, 4178837181, 8641346835
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 30 2021

Keywords

Crossrefs

Cf. A001764, A019497, A307971, A346733, A346735.

Programs

  • Mathematica
    nmax = 37; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 4, 1, Sum[Sum[a[i] a[j] a[n - i - j - 4], {j, 0, n - i - 4}], {i, 0, n - 4}]]; Table[a[n], {n, 0, 37}]

Formula

a(0) = ... = a(3) = 1; a(n) = Sum_{i=0..n-4} Sum_{j=0..n-i-4} a(i) * a(j) * a(n-i-j-4).
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