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.

Previous Showing 11-15 of 15 results.

A026634 a(n) = Sum_{k=0..floor(n/2)} A026626(n, k).

Original entry on oeis.org

1, 1, 4, 5, 15, 22, 59, 90, 230, 362, 902, 1450, 3551, 5802, 14022, 23210, 55492, 92842, 219974, 371370, 873101, 1485482, 3468893, 5941930, 13793183, 23767722, 54880915, 95070890, 218480607, 380283562, 870164852, 1521134250
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026626, A026627, A026628, A026629, A026630, A026631, A026632.
Cf. A026633, A026635, A026636, A026961, A026962, A026963, A026964.
Cf. A026965.

Programs

  • Magma
    b:= func< n | n le 2 select 2*n-1 else ((357*n^3-2696*n^2+6441*n-4822)*Self(n-1) +2*(2*n-7)*(51*n^2-203*n+188)*Self(n-2))/(2*(n-1)*(51*n^2-305*n+442)) >;
A026627:= [b(n+1) : n in [0..60]];
A026633:= [n le 1 select n+1 else (17*2^(n-2) +(-1)^n)/3 -1: n in [0..60]];
function A026634(n)
  if (n mod 2) eq 1 then return Floor(A026633[n+1]/2);
  else return Floor( (2*A026633[n+1] + (1+(-1)^n)*A026627[Floor(n/2) +1])/4);
  end if;
end function;
[A026634(n): n in [0..60]]; // G. C. Greubel, Jun 21 2024
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, (6*n-1 + (-1)^n)/4, T[n-1,k-1] +T[n-1,k]]];
A026634[n_]:= Sum[T[n,k], {k,0,n}];
Table[A026634[n], {n,0,40}] (* G. C. Greubel, Jun 21 2024 *)
  • SageMath
    @CachedFunction
def T(n, k): # T = A026626
    if (k==0 or k==n): return 1
    elif (k==1 or k==n-1): return int(3*n//2)
    else: return T(n-1, k-1) + T(n-1, k)
def A026634(n): return sum(T(n,k) for k in range((n//2)+1))
[A026634(n) for n in range(41)] # G. C. Greubel, Jun 21 2024

Formula

a(n) = floor(A026633(n)/2) if (n mod 2) = 1 and a(n) = floor((2*A026633(n) + (1+(-1)^n)*A026627(floor(n/2)+1))/4) if (n mod 2) = 0. - G. C. Greubel, Jun 21 2024

A026961 Self-convolution of array T given by A026626.

Original entry on oeis.org

1, 2, 11, 34, 138, 492, 1830, 6804, 25576, 96728, 367932, 1405884, 5392590, 20751504, 80076872, 309748096, 1200669828, 4662772672, 18137643524, 70657441212, 275620281310, 1076429623256, 4208562777342, 16470788108008, 64519534566362, 252948764993472, 992453764928050
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026626, A026627, A026628, A026629, A026630, A026631, A026632.
Cf. A026633, A026634, A026635, A026636, A026962, A026963, A026964.
Cf. A026965.

Programs

  • Magma
    p1:= func< n | -1864800 + 1239076*n + 7915984*n^2 - 11263411*n^3 + 5406551*n^4 - 1042185*n^5 + 65025*n^6 >;
p2:= func< n | -4505760 + 7236856*n + 10545958*n^2 - 20700889*n^3 + 10823147*n^4 - 2188767*n^5 + 143055*n^6 >;
p3:= func< n | -1522080 + 2667320*n + 3116288*n^2 - 6715322*n^3 + 3619972*n^4 - 755718*n^5 + 52020*n^6 >;
p4:= func< n | 42*(-376320 + 434044*n + 1225808*n^2 - 1997637*n^3 + 1002947*n^4 - 199767*n^5 + 13005*n^6) >;
p5:= func< n | 2*(-559440 + 1665230*n - 243157*n^2 - 1361078*n^3 + 898312*n^4 - 195432*n^5 + 13005*n^6) >;
I:=[11, 34, 138]; [1,2] cat [n le 3 select I[n] else (p1(n)*Self(n-1) + p2(n)*Self(n-2) + p3(n)*Self(n-3) + p4(n))/p5(n) : n in [1..40]]; // G. C. Greubel, Jun 21 2024
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, (6*n-1 + (-1)^n)/4, T[n-1,k-1] +T[n-1,k]]];
A026961[n_]:= A026961[n] = Sum[T[n,k]*T[n,n-k], {k,0,n}];
Table[A026961[n], {n,0,50}] (* G. C. Greubel, Jun 21 2024 *)
  • SageMath
    @CachedFunction
def T(n, k): # T = A026626
    if (k==0 or k==n): return 1
    elif (k==1 or k==n-1): return int(3*n//2)
    else: return T(n-1, k-1) + T(n-1, k)
def A026961(n): return sum(T(n,k)*T(n,n-k) for k in range(n+1))
[A026961(n) for n in range(41)] # G. C. Greubel, Jun 21 2024

Formula

From G. C. Greubel, Jun 21 2024: (Start)
a(n) = Sum_{k=0..n} T(n, k)*T(n, n-k). - G. C. Greubel, Jun 21 2024
a(n) = (p1(n)*a(n-1) + p2(n)*a(n-2) + p3(n)*a(n-3) + p4(n))/p5(n), where
p1(n) = 22589280 - 75610404*n + 85542748*n^2 - 44611965*n^3 + 11592851*n^4 - 1432335*n^5 + 65025*n^6.
p2(n) = 32659200 - 131052480*n + 161621002*n^2 - 88742247*n^3 + 23912807*n^4 - 3047097*n^5 + 143055*n^6.
p3(n) = 2*(5034960 - 21140910*n + 26659783*n^2 - 14896395*n^3 + 4089431*n^4 - 533919*n^5 + 26010*n^6).
p4(n) = 42*(3628800 - 13099136*n + 15429146*n^2 - 8267195*n^3 + 2196857*n^4 - 277797*n^5 + 13005*n^6).
p5(n) = 2*n*(-6580128 + 11379344*n - 7168746*n^2 + 2070547*n^3 - 273462*n^4 + 13005*n^5). (End)

Extensions

More terms from Sean A. Irvine, Oct 20 2019

A026962 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A026626.

Original entry on oeis.org

1, 6, 24, 108, 406, 1572, 5961, 22788, 87209, 335010, 1290376, 4983162, 19286891, 74797176, 290586771, 1130716508, 4406049037, 17191077082, 67152699384, 262594530318, 1027851765350, 4026831276662, 15788979175102, 61954847930374, 243278117470476, 955907159445522
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026626, A026627, A026628, A026629, A026630, A026631, A026632.
Cf. A026633, A026634, A026635, A026636, A026961, A026963, A026964.
Cf. A026965.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, Floor[3*n/2], T[n-1,k-1] +T[n-1,k]]]; (* T = A026626 *)
A262962[n_]:=Sum[T[n,k]*T[n,k+1], {k,0,n-1}];
Table[A262962[n], {n,40}] (* G. C. Greubel, Jun 23 2024 *)
  • SageMath
    @CachedFunction
def T(n, k): # T = A026626
    if (k==0 or k==n): return 1
    elif (k==1 or k==n-1): return int(3*n//2)
    else: return T(n-1, k-1) + T(n-1, k)
def A262962(n): return sum( T(n,k)*T(n,k+1) for k in range(n))
[A262962(n) for n in range(1,41)] # G. C. Greubel, Jun 23 2024

Extensions

More terms from Sean A. Irvine, Oct 20 2019

A026963 a(n) = Sum_{k=0..n-2} T(n,k) * T(n,k+2), with T given by A026626.

Original entry on oeis.org

1, 8, 52, 224, 987, 3980, 16057, 63732, 252424, 996332, 3927977, 15471622, 60915547, 239794516, 943946193, 3716205884, 14632901696, 57631689776, 227042423404, 894698122022, 3526753844436, 13906101471344, 54848887043366, 216402159510134, 854053133294062, 3371593602442500
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026626, A026627, A026628, A026629, A026630, A026631, A026632.
Cf. A026633, A026634, A026635, A026636, A026961, A026962, A026964.
Cf. A026965.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, Floor[3*n/2], T[n-1,k-1] +T[n-1,k]]]; (* T = A026626 *)
A262963[n_]:= Sum[T[n,k]*T[n,k+2], {k,0,n-2}];
Table[A262963[n], {n,2,40}] (* G. C. Greubel, Jun 23 2024 *)
  • SageMath
    @CachedFunction
def T(n, k): # T = A026626
    if (k==0 or k==n): return 1
    elif (k==1 or k==n-1): return int(3*n//2)
    else: return T(n-1, k-1) + T(n-1, k)
def A262963(n): return sum( T(n,k)*T(n,k+2) for k in range(n-1))
[A262963(n) for n in range(2,41)] # G. C. Greubel, Jun 23 2024

Extensions

More terms from Sean A. Irvine, Oct 20 2019

A026965 a(n) = Sum_{k=0..n} (k+1) * A026626(n,k).

Original entry on oeis.org

1, 3, 10, 25, 66, 154, 360, 810, 1810, 3982, 8700, 18850, 40614, 87030, 185680, 394570, 835578, 1763998, 3713700, 7798770, 16340302, 34166086, 71303160, 148548250, 308980386, 641728494, 1330992460, 2757055810, 5704253430
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026626, A026627, A026628, A026629, A026630, A026631, A026632.
Cf. A026633, A026634, A026635, A026636, A026961, A026962, A026963.
Cf. A026964.

Programs

  • Magma
    [n le 1 select 2*n+1 else (n+2)*(17*2^(n-2) -3 +(-1)^n)/6: n in [0..40]]; // G. C. Greubel, Jun 23 2024
  • Mathematica
    Table[(n+2)*(17*2^(n-2) -3 +(-1)^n)/6 +(1/4)*(Boole[n==0] +3*Boole[n== 1]), {n,0,50}] (* G. C. Greubel, Jun 23 2024 *)
  • SageMath
    [(n+2)*(17*2^(n-2) -3 +(-1)^n)/6 + (1/4)*(int(n==0) + 3*int(n==1)) for n in range(41)] # G. C. Greubel, Jun 23 2024

Formula

G.f.: (1-x-x^3+3*x^4-5*x^5-2*x^6+3*x^7)/((1-x)^2*(1+x)^2*(1-2*x)^2). - Colin Barker, Apr 26 2015
From G. C. Greubel, Jun 23 2024: (Start)
a(n) = (1/6)*(n+2)*(17*2^(n-2) - 3 + (-1)^n) + (1/4)*([n=0] + 3*[n=1]).
a(n) = ( n*(n+2)*a(n-1) + 2*(n+1)*(n+2)*a(n-2) + n*(n+1)*(n+2) )/(n*(n + 1)), with a(0) = 1, a(1) = 3, a(2) = 10, a(3) = 25.
E.g.f.: (1/12)*( 17*(1+x)*exp(2*x) - 6*(2+x)*exp(x) + 2*(2-x)*exp(-x) + 3*(1+3*x) ). (End)
Previous Showing 11-15 of 15 results.

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