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
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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
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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 *)
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@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
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
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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
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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 *)
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@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
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
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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 *)
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@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
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
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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 *)
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@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
A026964
a(n) = Sum_{k=0..n-3} T(n,k) * T(n,k+3), with T given by A026626.
Original entry on oeis.org
1, 12, 77, 434, 1978, 8830, 37409, 156474, 644305, 2632506, 10684360, 43166246, 173768764, 697596990, 2794438513, 11174809302, 44626341136, 178018744896, 709505830530, 2825762505810, 11247704919634, 44749537493028, 177970696795672, 707580176408854, 2812524327414647
Offset: 3
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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 *)
A262964[n_]:= Sum[T[n,k]*T[n,k+3], {k,0,n-3}];
Table[A262964[n], {n,3,40}] (* G. C. Greubel, Jun 23 2024 *)
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@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 A262964(n): return sum( T(n,k)*T(n,k+3) for k in range(n-2))
[A262964(n) for n in range(3,41)] # G. C. Greubel, Jun 23 2024