A026725 -id:A026725 - cp's OEIS frontend

A026844 a(n) = T(2n,n+4), T given by A026725.

1, 10, 67, 379, 1958, 9576, 45190, 208084, 941480, 4204949, 18597694, 81635060, 356220369, 1547066801, 6693361973, 28868868733, 124194904215, 533156609953, 2284722747583, 9776008778375, 41777089615201, 178338353574365, 760586650190997

Offset: 4

Clark Kimberling

nonn

Column k=9 of triangle A236830. - Philippe Deléham, Feb 02 2014

G. C. Greubel, Table of n, a(n) for n = 4..1000

Cf. A000108, A026725, A236830.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-4*x))^9/(64*x^3*(8*x^2 -(1-Sqrt(1-4*x))^3)) )); // G. C. Greubel, Jul 19 2019

Drop[CoefficientList[Series[(1-Sqrt[1-4x])^9/(64*x^3*(8*x^2-(1-Sqrt[1-4x])^3)), {x,0,40}], x], 4] (* G. C. Greubel, Jul 19 2019 *)

my(x='x+O('x^40)); Vec( (1-sqrt(1-4*x))^9/(64*x^3*(8*x^2 -(1-sqrt(1-4*x))^3)) ) \\ G. C. Greubel, Jul 19 2019

a=((1-sqrt(1-4*x))^9/(64*x^3*(8*x^2 -(1-sqrt(1-4*x))^3)) ).series(x, 45).coefficients(x, sparse=False); a[4:40] # G. C. Greubel, Jul 19 2019

G.f.: (x^4*C(x)^9)/(1-x*C(x)^3) where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014
a(n) ~ phi^(3*n-5) / sqrt(5), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 19 2019
(7719*n+49402)*(n+3)*a(n) +(7719*n^2-1057949*n-1942990)*a(n-1) +4*(-211672*n^2+2076533*n+763807)*a(n-2) +(4326581*n^2-34087269*n+38502298)*a(n-3) +3*(-1940897*n^2+16395555*n-37085206)*a(n-4) -2*(406705*n-1575734)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Oct 26 2019

A026734 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026725.

Original entry on oeis.org

1, 3, 7, 16, 34, 74, 154, 330, 682, 1451, 2989, 6332, 13018, 27495, 56449, 118954, 243964, 513180, 1051612, 2208856, 4523344, 9489604, 19422124, 40704746, 83269990, 174366100, 356558320, 746073604, 1525104172, 3189119418

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Partial sums of A026732.

T:= function(n,k)
    if n<0 then return 0;
    elif k=0 or k=n then return 1;
    elif 2*k=n-1 then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
    else return T(n-1, k-1) + T(n-1, k);
    fi;
  end;
List([0..30], n-> Sum([0..n], k-> Sum([0..n], j-> T(j,k) )))); # G. C. Greubel, Oct 26 2019

A026725:= proc(n,k) option remember;
    if n<0 or k<0 then 0;
    elif k=0 or k=n then 1;
    elif 2*k = n-1 then procname(n-1,k-1)+procname(n-2,k-1) + procname(n-1,k) ;
   else procname(n-1,k-1)+procname(n-1,k) ;
fi; end proc:seq(add(add(A026725(i,j), j=0..n), i=0..n), n=0..30); # G. C. Greubel, Oct 26 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k==(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]]]]; Table[Sum[T[j, k], {k,0,n}, {j,0,n}], {n,0,30}] (* G. C. Greubel, Oct 26 2019 *)

T(n,k) = if(n<0, 0, if(k==n || k==0, 1, if(2*k==n-1, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) )));
vector(31, n, sum(j=0,n-1, sum(i=0,n-1, T(j,i))) ) \\ G. C. Greubel, Oct 26 2019

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k==(n-1)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[sum( sum( T(j, k) for k in (0..n)) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Oct 26 2019

A026735 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026725.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 15, 28, 43, 71, 130, 201, 331, 597, 928, 1525, 2720, 4245, 6965, 12315, 19280, 31595, 55472, 87067, 142539, 248802, 391341, 640143, 1111864, 1752007, 2863871, 4953162, 7817033, 12770195, 22004810, 34775005, 56779815

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

T:= function(n,k)
    if n<0 then return 0;
    elif k=0 or k=n then return 1;
    elif 2*k=n-1 then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
    else return T(n-1, k-1) + T(n-1, k);
    fi;
  end;
List([0..30], n-> Sum([0..Int(n/2)], k-> T(n-k,k) )); # G. C. Greubel, Oct 26 2019

A026725:= proc(n,k) option remember;
    if n<0 or k<0 then 0;
    elif k=0 or k=n then 1;
    elif 2*k = n-1 then procname(n-1,k-1)+procname(n-2,k-1) + procname(n-1,k) ;
   else procname(n-1,k-1)+procname(n-1,k) ;
   fi;
end proc:
seq(add(A026725(n-k,k), k=0..floor(n/2)), n=0..30); # G. C. Greubel, Oct 26 2019

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k==(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]]]]; Table[Sum[T[n-k, k], {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Oct 26 2019 *)

T(n,k) = if(n<0, 0, if(k==n || k==0, 1, if(2*k==n-1, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) )));
vector(31, n, sum(j=0,(n-1)\2, T(n-j,j)) ) \\ G. C. Greubel, Oct 26 2019

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k==(n-1)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[sum( T(n-j, j) for j in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 26 2019

Conjecture: G.f.:-1/2*(2*x^6-5*x^4+8*x^3+x-2+x*(x-1)*(x^2+x+1)*(1-4*x^3)^(1/2))/(x^6+4*x^3-1)/(x^2+x-1). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

A026847 a(n) = T(n,m) + T(n,m+1) + ... + T(n,n), where m=[ (n+1)/2 ], T given by A026725.

Original entry on oeis.org

1, 1, 3, 4, 12, 17, 50, 73, 211, 314, 895, 1350, 3805, 5798, 16193, 24872, 68940, 106573, 293526, 456169, 1249622, 1950697, 5318976, 8334539, 22634700, 35582783, 96296410, 151809737, 409573584, 647279131, 1741574006, 2758310121

Offset: 0

Clark Kimberling

nonn

A027207 Self-convolution of row n of array T given by A026725.

Original entry on oeis.org

1, 2, 6, 26, 91, 414, 1519, 6980, 26176, 120596, 457880, 2110422, 8076044, 37211408, 143165518, 659264978, 2546261659, 11717539646, 45387607779, 208729977124, 810316893827

Offset: 0

Clark Kimberling

nonn

A027207 := proc(n)
    add( A026725(n,k)*A026725(n,n-k),k=0..n) ;
end proc: # R. J. Mathar, Oct 21 2019

A027208 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A026725.

Original entry on oeis.org

1, 4, 19, 72, 338, 1289, 6027, 23117, 107746, 415058, 1929383, 7456777, 34584731, 134005510, 620344059, 2408498148, 11131638368, 43289074671, 199800878749, 778025282080, 3586777011141, 13982303458588, 64394916131100, 251261623174835, 1156165698462868, 4514741175877259

Offset: 1

Clark Kimberling

nonn

A027208 := proc(n)
    add( A026725(n,k)*A026725(n,k+1),k=0..n-1) ;
end proc: # R. J. Mathar, Oct 21 2019

Name corrected by R. J. Mathar, Oct 21 2019

More terms from Sean A. Irvine, Oct 24 2019

A027209 a(n) = Sum_{k=0..n-2} T(n,k) * T(n,k+2), with T given by A026725.

Original entry on oeis.org

1, 7, 34, 173, 741, 3549, 14670, 68683, 279715, 1294598, 5233060, 24052965, 96822587, 442972139, 1778697948, 8110611295, 32516501991, 147897243768, 592346459071, 2688895275974, 10762289869782, 48776681304283, 195143578562778, 883272843800993, 3532739195660162

Offset: 2

Clark Kimberling

nonn

A027209 := proc(n)
    add( A026725(n,k)*A026725(n,k+2),k=0..n-2) ;
end proc: # R. J. Mathar, Oct 21 2019

Name corrected by R. J. Mathar, Oct 21 2019

More terms from Sean A. Irvine, Oct 24 2019

A027210 a(n) = Sum_{k=0..n-3} T(n,k) * T(n,k+3), with T given by A026725.

Original entry on oeis.org

1, 9, 57, 298, 1545, 7126, 34603, 151633, 714481, 3049328, 14131926, 59385159, 272409611, 1133433579, 5163641033, 21341122875, 96751092786, 397978832144, 1797722306693, 7369285976135, 33195299823112, 135723491400062, 610027809595308, 2489247495517382, 11168400384253936

Offset: 3

Clark Kimberling

nonn

A027210 := proc(n)
    add( A026725(n,k)*A026725(n,k+3),k=0..n-3) ;
end proc: # R. J. Mathar, Oct 21 2019

Name corrected by R. J. Mathar, Oct 21 2019

More terms from Sean A. Irvine, Oct 24 2019

A027211 a(n) = Sum_{k=0..n} (k+1) * A026725(n, k).

Original entry on oeis.org

1, 3, 8, 22, 53, 136, 312, 768, 1712, 4101, 8971, 21082, 45507, 105435, 225347, 516424, 1095353, 2488480, 5246176, 11834584, 24826412, 55677436, 116321132, 259585768, 540454158, 1201026646, 2493149402, 5520322840, 11430160964

Offset: 0

Clark Kimberling

nonn

A027211 := proc(n)
    add( (k+1)*A026725(n,k),k=0..n) ;
end proc: # R. J. Mathar, Oct 21 2019

Name corrected by R. J. Mathar, Oct 21 2019

A027212 a(n) = Sum_{k=0..n} (k+1) * A026725(n, n-k).

Original entry on oeis.org

1, 3, 8, 23, 55, 144, 328, 816, 1808, 4358, 9485, 22377, 48097, 111720, 237917, 546161, 1154827, 2626624, 5522464, 12467540, 26092324, 58546544, 122059348, 272479782, 566242186, 1258568324, 2608232758, 5775620396, 11940756076

Offset: 0

Clark Kimberling

nonn

A027212 := proc(n)
    add( (k+1)*A026725(n,n-k),k=0..n) ;
end proc: # R. J. Mathar, Oct 21 2019

Name corrected by R. J. Mathar, Oct 21 2019

cp's OEIS Frontend

A026844 a(n) = T(2n,n+4), T given by A026725.

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A026734 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026725.

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A026735 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026725.

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A026847 a(n) = T(n,m) + T(n,m+1) + ... + T(n,n), where m=[ (n+1)/2 ], T given by A026725.

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A027207 Self-convolution of row n of array T given by A026725.

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Maple

A027208 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A026725.

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A027209 a(n) = Sum_{k=0..n-2} T(n,k) * T(n,k+2), with T given by A026725.

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A027210 a(n) = Sum_{k=0..n-3} T(n,k) * T(n,k+3), with T given by A026725.

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A027211 a(n) = Sum_{k=0..n} (k+1) * A026725(n, k).

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Maple

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A027212 a(n) = Sum_{k=0..n} (k+1) * A026725(n, n-k).

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