A027052 -id:A027052 - cp's OEIS frontend

A027074 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,2n-k), with T given by A027052.

1, 1, 4, 22, 93, 389, 1570, 6144, 23629, 89551, 335430, 1244762, 4583293, 16765087, 60980096, 220724896, 795540601, 2856541663, 10222762962, 36475315442, 129796579409, 460757642587, 1632012075912, 5768986242408

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( add(T(n,k)*T(n,2*n-k), k=0..n-1), n=1..30); # G. C. Greubel, Nov 06 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[Sum[T[n,k]*T[n,2*n-k], {k,0,n-1}], {n, 30}] (* G. C. Greubel, Nov 06 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum(T(n,k)*T(n,2*n-k) for k in (0..n-1)) for n in (1..30)] # G. C. Greubel, Nov 06 2019

A027075 a(n) = Sum_{k=0..2n} (k+1) * A027052(n, k).

Original entry on oeis.org

1, 4, 17, 58, 199, 682, 2301, 7654, 25145, 81740, 263407, 842720, 2679935, 8479378, 26713555, 83847748, 262335577, 818473148, 2547289679, 7910433568, 24517303535, 75854736178, 234317624167, 722776320072, 2226565995913

Offset: 0

Clark Kimberling

nonn

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

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( add((k+1)*T(n,k), k=0..2*n), n=0..30); # G. C. Greubel, Nov 06 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[Sum[(k+1)*T[n,k], {k, 0, 2*n}], {n,0,30}] (* G. C. Greubel, Nov 06 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum((k+1)*T(n,k) for k in (0..2*n)) for n in (0..30)] # G. C. Greubel, Nov 06 2019

A027076 a(n) = Sum_{k=0..2n} (k+1) * A027052(n, 2n-k).

Original entry on oeis.org

1, 4, 13, 38, 111, 326, 961, 2842, 8425, 25020, 74403, 221488, 659895, 1967422, 5869055, 17516540, 52300729, 156214828, 466736979, 1394894672, 4169810935, 12467680862, 37285474803, 111524444760, 333633526937, 998233861836

Offset: 0

Clark Kimberling

nonn

The terms a(0)..a(25) obey a linear recurrence with polynomial coefficients of degree 7. - Ralf Stephan, May 31 2014

			G.f. = 1 + 4*x + 13*x^2 + 38*x^3 + 111*x^4 + 326*x^5 + 961*x^6 + 2842*x^7 + ...

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

Cf. A027052.

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( add((k+1)*T(n,2*n-k), k=0..2*n), n=0..30); # G. C. Greubel, Nov 06 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[Sum[(k+1)*T[n,2*n-k], {k, 0, 2*n}], {n, 0, 30}] (* G. C. Greubel, Nov 06 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum((k+1)*T(n,2*n-k) for k in (0..2*n)) for n in (0..30)] # G. C. Greubel, Nov 06 2019

0 = a(n)*(9*n + 9) + a(n+1)*(3*n + 21) + a(n+2)*(13*n - 5) + a(n+3)*(-29*n + 11) + a(n+4)*(-13*n - 121) + a(n+5)*(25*n + 123) + a(n+6)*(-98n - 43) + a(n+7)*(n + 5) for n>=-1. - Michael Somos, May 31 2014

0 = a(n)*(+81*a(n+1) + 189*a(n+2) + ... + 45*a(n+8)) + a(n+1)*(-135*a(n+1) + ...) + ... + a(n+7)*(-7*a(n+7) + a(n+8)) for n>=-1. - Michael Somos, May 31 2014

A027077 a(n) = Sum_{k=n+1..2*n} T(n,k), T given by A027052.

Original entry on oeis.org

1, 3, 8, 24, 71, 209, 612, 1784, 5189, 15081, 43838, 127528, 371395, 1082951, 3161866, 9243400, 27055153, 79280601, 232567194, 682905120, 2007104343, 5904004451, 17380510458, 51202600920, 150942696637, 445247984543

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( add(T(n,k), k=n+1..2*n), n=1..30); # G. C. Greubel, Nov 06 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[Sum[T[n, k], {k, n+1, 2*n}], {n, 1, 30}] (* G. C. Greubel, Nov 06 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum(T(n,k) for k in (n+1..2*n)) for n in (1..30)] # G. C. Greubel, Nov 06 2019

Offset changed by G. C. Greubel, Nov 06 2019

A027078 a(n) = Sum_{k=0..n} T(n,k) * T(n,n+k), with T given by A027052.

Original entry on oeis.org

1, 0, 2, 8, 31, 130, 590, 2798, 13541, 66724, 332708, 1673536, 8479367, 43218034, 221383712, 1138976166, 5882112985, 30479772624, 158413903096, 825556260636, 4312814257059, 22580855859166, 118468635595680, 622698941708890

Offset: 0

Clark Kimberling

nonn

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

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( add(T(n,k)*T(n,n+k), k=0..n), n=0..30); # G. C. Greubel, Nov 07 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[Sum[T[n, k]*T[n, n+k], {k, 0, n}], {n,30}] (* G. C. Greubel, Nov 07 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum(T(n,k)*T(n,n+k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 07 2019

A027079 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A027052.

Original entry on oeis.org

0, 4, 24, 160, 1136, 8420, 64224, 499984, 3952928, 31634724, 255682432, 2083562368, 17097573344, 141143273396, 1171240794072, 9763809318912, 81724975129664, 686539343850164

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( add(T(n,k)*T(n,k+1), k=0..2*n-1), n=1..30); # G. C. Greubel, Nov 07 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[Sum[T[n, k]*T[n, k+1], {k, 0, 2*n-1}], {n,30}] (* G. C. Greubel, Nov 07 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum(T(n,k)*T(n,k+1) for k in (0..2*n-1)) for n in (1..30)] # G. C. Greubel, Nov 07 2019

A027080 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A027052.

Original entry on oeis.org

2, 15, 100, 757, 5902, 46907, 377520, 3065809, 25078650, 206416795, 1708129244, 14202265321, 118585167502, 993915161547, 8358970631568, 70518298143329, 596590060985546, 5060232622624651, 43022268222676124, 366575545244139845, 3129747701356459022, 26771150349554898415

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( add(T(n,k)*T(n,k+2), k=0..2*n-2), n=2..30); # G. C. Greubel, Nov 07 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[Sum[T[n, k]*T[n, k+2], {k, 0, 2*n-2}], {n,2,30}] (* G. C. Greubel, Nov 07 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum(T(n,k)*T(n,k+2) for k in (0..2*n-2)) for n in (2..30)] # G. C. Greubel, Nov 07 2019

More terms from Sean A. Irvine, Oct 22 2019

A027081 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A027052.

Original entry on oeis.org

8, 56, 438, 3574, 29738, 249200, 2094902, 17648718, 148968822, 1259807224, 10674450652, 90618393250, 770728674864, 6567151658496, 56054864624310, 479267092351534, 4104271159315190, 35200977081482376, 302343415930398696, 2600408469332918538, 22394817457275426524

Offset: 3

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 3..1000

T:= proc(n, k) option remember;
      if k<0 or k>2*n then 0
    elif k=0 or k=2 or k=2*n then 1
    elif k=1 then 0
    else add(T(n-1, k-j), j=1..3)
      fi
    end:
seq( add(T(n,k)*T(n,k+3), k=0..2*n-3), n=3..30); # G. C. Greubel, Nov 07 2019

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[Sum[T[n, k]*T[n, k+3], {k, 0, 2*n-3}], {n,3,30}] (* G. C. Greubel, Nov 07 2019 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum(T(n,k)*T(n,k+3) for k in (0..2*n-3)) for n in (3..30)] # G. C. Greubel, Nov 07 2019

More terms from Sean A. Irvine, Oct 22 2019

A160999 Row sums of A027052.

Original entry on oeis.org

1, 2, 5, 12, 31, 84, 233, 656, 1865, 5338, 15355, 44342, 128455, 373100, 1086087, 3167634, 9254009, 27074666, 79316491, 232633206, 683026535, 2007327660, 5904415195, 17381265934, 51203990457, 150945252394, 445252685313

Offset: 0

R. J. Mathar, Jun 01 2009

			a(2) = 1+0+1+2+1 = 5.
a(3) = 1+0+1+2+3+4+1 = 12.

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

A027052 := proc(n,k) option remember; if k =0 or k = 2*n then 1; elif k = 1 then 0; elif k =2 then 1; else procname(n-1,k-3)+procname(n-1,k-2)+procname(n-1,k-1) ; fi; end:
A160999 := proc(n) add( A027052(n,k),k=0..2*n) ; end: seq(A160999(n),n=0..30) ;

T[n_, k_]:= T[n, k]= If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]; Table[Sum[T[n, k], {k,0,2*n}], {n,0,30}] (* G. C. Greubel, Nov 06 2019 *)

@CachedFunction
def T(n, k):
    if (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum(T(n, k) for k in (0..2*n)) for n in (0..30)] # G. C. Greubel, Nov 06 2019

a(n) = Sum_{k=0..2*n} A027052(n,k).

Conjecture: (-n+2)*a(n) +(6*n-11)*a(n-1) +(-7*n+1)*a(n-2) +2*(-4*n+27)*a(n-3) +(5*n-28)*a(n-4) +(2*n-3)*a(n-5) +3*(n-5)*a(n-6)=0. - R. J. Mathar, May 26 2016

A027071 Greatest number in row n of array T given by A027052.

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 59, 153, 406, 1126, 3124, 8684, 24202, 67640, 189576, 532786, 1519151, 4356471, 12501301, 35901325, 103188123, 296844379, 854701935, 2463133311, 7104685935, 20510632575, 59262772629, 172271893036, 502157965938

Offset: 0

Clark Kimberling

nonn

cp's OEIS Frontend

A027074 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,2n-k), with T given by A027052.

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A027075 a(n) = Sum_{k=0..2n} (k+1) * A027052(n, k).

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A027076 a(n) = Sum_{k=0..2n} (k+1) * A027052(n, 2n-k).

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A027077 a(n) = Sum_{k=n+1..2*n} T(n,k), T given by A027052.

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A027078 a(n) = Sum_{k=0..n} T(n,k) * T(n,n+k), with T given by A027052.

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A027079 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A027052.

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Maple

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Sage

A027080 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A027052.

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A027081 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A027052.

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