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.

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A027103 a(n) = self-convolution of row n of array T given by A027082.

Original entry on oeis.org

1, 3, 11, 41, 161, 633, 2457, 9415, 35663, 133721, 497025, 1833499, 6719707, 24488183, 88798775, 320596441, 1152996137, 4132351481, 14764629473, 52606224267, 186963146315, 662947211163, 2345810793755, 8284654896229
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027104 a(n) = Sum_{k=0..n} T(n,k) * T(n,2n-k), with T given by A027082.

Original entry on oeis.org

1, 2, 6, 25, 93, 357, 1373, 5188, 19456, 72373, 267137, 979762, 3573058, 12965292, 46839228, 168552205, 604420933, 2160638253, 7701879217, 27384190634, 97138835970, 343846037094, 1214761031998, 4283924048755
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027105 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,2n-k), with T given by A027082.

Original entry on oeis.org

1, 5, 16, 68, 276, 1084, 4227, 16207, 61348, 229888, 853737, 3146649, 11522891, 41959547, 152044236, 548575204, 1971713228, 7062750256, 25222033633, 89824310345, 319101174069, 1131049761757, 4000730847474, 14124602551114
Offset: 1

Views

Author

Clark Kimberling

Keywords

A027106 a(n) = Sum_{k=0..2n} (k+1) * A027082(n, k).

Original entry on oeis.org

1, 6, 28, 114, 422, 1498, 5146, 17282, 57088, 186234, 601618, 1928314, 6141164, 19453974, 61349680, 192729074, 603449486, 1883989018, 5866941586, 18229205738, 56526585004, 174967361838, 540701808500, 1668482208274
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027108 a(n) = T(n,n+1) + T(n,n+2) + ... + T(n,2n), T given by A027082.

Original entry on oeis.org

1, 5, 16, 50, 152, 452, 1331, 3897, 11378, 33186, 96791, 282461, 825013, 2412157, 7060154, 20686324, 60673102, 178126774, 523426827, 1539393889, 4530901919, 13345545959, 39335284312, 116011722730, 342354698603
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027109 a(n) = Sum_{k=0..n} T(n,k) * T(n,n+k), with T given by A027082.

Original entry on oeis.org

1, 2, 6, 29, 131, 619, 3013, 14854, 73996, 372023, 1883369, 9591012, 49091034, 252371780, 1302419484, 6744389329, 35031370911, 182457629783, 952674694705, 4985494458892, 26143834054498, 137358293523986, 722938114992478
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A027082.

Programs

  • Mathematica
    T[, 0] = T[, 1] = T[_, 2] = 1;
T[n_, k_] := T[n, k] = If[k < 2n, T[n-1, k-3] + T[n-1, k-2] + T[n-1, k-1], T[n-1, 2n-3] + T[n-1, 2n-2]];
a[n_] := Sum[T[n, k] T[n, n+k], {k, 0, n}];
a /@ Range[0, 22] (* Jean-François Alcover, Sep 26 2020 *)

A027110 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A027082.

Original entry on oeis.org

2, 11, 80, 591, 4492, 34859, 274834, 2194175, 17697088, 143951359, 1179361946, 9721994363, 80573155210, 670916858663, 5609929650932, 47082821529795, 396479220532944, 3348822504340655, 28363386616346538, 240832364304754043, 2049614659600342062
Offset: 1

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 23 2019

A027111 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A027082.

Original entry on oeis.org

6, 52, 404, 3237, 26086, 211730, 1730114, 14223429, 117563545, 976408938, 8144436109, 68197885773, 573056586975, 4830577196508, 40836850965484, 346138265331977, 2941011994449017, 25044401420362382, 213706426365981010, 1827057028991860141, 15647888579758708485
Offset: 2

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 23 2019

A027112 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A027082.

Original entry on oeis.org

29, 238, 2012, 16975, 143106, 1207017, 10191450, 86181211, 730023605, 6195099626, 52668845006, 448581627313, 3827264611569, 32708905417122, 279989900778562, 2400407089060955, 20609200162062499, 177190108342853474, 1525420499563895994, 13148693465859069309
Offset: 3

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 23 2019

A027052 Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0, T(n,1)=0 for n >= 1, T(n,2)=1 for n >= 2 and for n >= 3, T(n,k) = T(n-1,k-3) + T(n-1, k-2) + T(n-1,k-1) for 3 <= k <= 2n-1. T(n,k)=0 for k < 0 or k > 2n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 3, 4, 1, 1, 0, 1, 2, 3, 6, 9, 8, 1, 1, 0, 1, 2, 3, 6, 11, 18, 23, 18, 1, 1, 0, 1, 2, 3, 6, 11, 20, 35, 52, 59, 42, 1, 1, 0, 1, 2, 3, 6, 11, 20, 37, 66, 107, 146, 153, 102, 1, 1, 0, 1, 2, 3, 6, 11, 20, 37, 68, 123, 210, 319, 406, 401, 256, 1
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003

Examples

			Triangle T(n,k) for 0 <= k <= 2n:
  1;
  1, 0, 1;
  1, 0, 1, 2, 1;
  1, 0, 1, 2, 3, 4, 1;
  1, 0, 1, 2, 3, 6, 9, 8, 1;

Links

Crossrefs

Cf. A001590, a tribonacci sequence.
Cf. A160999 (row sums), A005408 (row lengths).
Diagonals T(n, n+c): A027053 (c=2), A027054 (c=3), A027055 (c=4).
Diagonals T(n, 2n-c): A027056 (c=1), A027058 (c=2), A027059 (c=3), A027060 (c=4), A027061(c=5), A027062 (c=6), A027063 (c=7), A027064 (c=8), A027065 (c=9), A027066 (c=10).
Sums involving this array: A027067, A027068, A027069, A027070, A027072, A027073, A027074, A027075, A027076, A027077,A027078, A027079, A027080, A027081.
Other related sequences: A027057, A027071.
Other arrays of this type: A027023, A027082, A027113.

Programs

  • GAP
    T:= function(n,k)
    if k=0 or k=2 or k=2*n then return 1;
    elif k=1 then return 0;
    else return Sum([1..3], j-> T(n-1, k-j) );
    fi;
  end;
Flat(List([0..10], n-> List([0..2*n], k-> T(n,k) ))); # G. C. Greubel, Nov 05 2019
  • Maple
    T:= proc(n, k) option remember;
      if 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(seq(T(n, k), k=0..2*n), n=0..10); # G. C. Greubel, Nov 05 2019
  • Mathematica
    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[T[n, k], {n, 0, 12}, {k, 0, 2*n}]//Flatten (* G. C. Greubel, Nov 05 2019 *)
  • PARI
    {T(n, k) = if(k==0 || k==2 || k==2*n, 1, if(k==1, 0, sum(j=1,3, T(n-1, k-j)) ))};
for(n=0, 10, for(k=0,2*n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 05 2019
  • Sage
    @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))
[[T(n, k) for k in (0..2*n)] for n in (0..10)] # G. C. Greubel, Nov 05 2019

Formula

A001590(k+1) = T(n,k) if 0 <= k <= n. - Michael Somos, Jun 01 2014

Extensions

Offset and keyword:tabl corrected by R. J. Mathar, Jun 01 2009
Previous Showing 21-30 of 32 results. Next

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