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.

Showing 1-7 of 7 results.

A055818 Triangle T read by rows: T(i,j) = R(i-j,j), where R(i,0) = R(0,i) = 1 for i >= 0, R(i,j) = Sum_{h=0..i-1} Sum_{m=0..j} R(h,m) for i >= 1, j >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 11, 9, 4, 1, 1, 23, 24, 14, 5, 1, 1, 47, 60, 43, 20, 6, 1, 1, 95, 144, 122, 69, 27, 7, 1, 1, 191, 336, 328, 217, 103, 35, 8, 1, 1, 383, 768, 848, 640, 354, 146, 44, 9, 1, 1, 767, 1728, 2128, 1800, 1131, 543, 199, 54, 10, 1
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Examples

			Rows begins as:
  1;
  1,  1;
  1,  2, 1;
  1,  5, 3, 1;
  1, 11, 9, 4, 1;
  ...

Links

Crossrefs

Cf. A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055826, A055827, A055828, A055829.

Programs

  • GAP
    T:= function(i,j)
    if i=0 or j=0 then return 1;
    else return Sum([0..i-1], h-> Sum([0..j], m-> T(h,m) ));
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n-k,k) ))); # G. C. Greubel, Jan 21 2020
  • Magma
    function T(i,j)
  if i eq 0 or j eq 0 then return 1;
  else return (&+[(&+[T(h,m): m in [0..j]]): h in [0..i-1]]);
  end if; return T; end function;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 21 2020
  • Maple
    T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
      fi; end:
seq(seq(T(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Jan 21 2020
  • Mathematica
    T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 21 2020 *)
  • PARI
    T(i,j) = if(i==0 || j==0, 1, sum(h=0,i-1, sum(m=0,j, T(h,m) )));
for(n=0,12, for(k=0, n, print1(T(n-k,k), ", "))) \\ G. C. Greubel, Jan 21 2020
  • Sage
    @CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[[T(n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 21 2020

A055823 a(n) = T(n,n-6), array T as in A055818.

Original entry on oeis.org

1, 95, 336, 848, 1800, 3422, 6017, 9974, 15782, 24045, 35498, 51024, 71672, 98676, 133475, 177734, 233366, 302555, 387780, 491840, 617880, 769418, 950373, 1165094, 1418390, 1715561, 2062430, 2465376, 2931368, 3468000, 4083527, 4786902, 5587814, 6496727, 7524920
Offset: 6

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818, A055819, A055820, A055821, A055822, A055824, A055825, A055826, A055827, A055828, A055829.

Programs

  • GAP
    Concatenation([1], List([7..50], n-> (n^6 +15*n^5 -65*n^4 -795*n^3 +1864*n^2 +6180*n -7200)/720 )); # G. C. Greubel, Jan 22 2020
  • Magma
    I:=[1,95,336,848,1800,3422,6017,9974,15782,24045, 35498,51024,71672]; [n le 13 select I[n] else 7*Self(n-1)- 21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..50]]; // Vincenzo Librandi, Dec 30 2016
  • Maple
    seq( `if`(n=6, 1, (n^6 +15*n^5 -65*n^4 -795*n^3 +1864*n^2 +6180*n -7200)/720), n=6..50); # G. C. Greubel, Jan 22 2020
  • Mathematica
    Join[{1, 95, 336, 848, 1800, 3422}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {6017, 9974, 15782, 24045, 35498, 51024, 71672}, 50]] (* Vincenzo Librandi, Dec 30 2016 *)
Table[If[n==6, 1, (n^6 +15*n^5 -65*n^4 -795*n^3 +1864*n^2 +6180*n -7200)/720], {n, 6, 50}] (* G. C. Greubel, Jan 22 2020 *)
  • PARI
    vector(45, n, my(m=n+5); if(m==6, 1, (m^6 +15*m^5 -65*m^4 -795*m^3 +1864*m^2 +6180*m -7200)/720)) \\ G. C. Greubel, Jan 22 2020
  • Sage
    [1]+[(n^6 +15*n^5 -65*n^4 -795*n^3 +1864*n^2 +6180*n -7200)/720 for n in (7..50)] # G. C. Greubel, Jan 22 2020

Formula

From Chai Wah Wu, Dec 29 2016: (Start)
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 13.
G.f.: x^6*(1 + 88*x - 308*x^2 + 456*x^3 - 370*x^4 + 174*x^5 - 45*x^6 + 5*x^7)/(1-x)^7. (End)
From G. C. Greubel, Jan 22 2020: (Start)
a(n) = (n^6 + 15*n^5 - 65*n^4 - 795*n^3 + 1864*n^2 + 6180*n -7200)/720, for n > 6, with a(6) = 1.
E.g.f.: (7200 - 2880*x^2 - 960*x^3 + 30*x^4 + 60*x^5 - 5*x^6 + (-7200 + 7200*x - 720*x^2 - 720*x^3 + 150*x^4 + 30*x^5 + x^6)*exp(x))/720. (End)

A055824 a(n) = T(2*n,n), array T as in A055818.

Original entry on oeis.org

1, 2, 9, 43, 217, 1131, 6017, 32467, 177009, 972691, 5378425, 29889531, 166795977, 934039419, 5246059761, 29540072355, 166708076001, 942651407907, 5339465635049, 30291114653131, 172081678284729, 978807205953931
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055825, A055826, A055827, A055828, A055829.

Programs

  • Maple
    T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
fi; end: seq(T(n, n), n=0..30); # G. C. Greubel, Jan 22 2020
  • Mathematica
    T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)
  • Sage
    @CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055825 a(n) = T(2n+1,n), array T as in A055818.

Original entry on oeis.org

1, 5, 24, 122, 640, 3422, 18536, 101362, 558336, 3093302, 17218168, 96214890, 539415552, 3032659086, 17091411912, 96527966178, 546184965120, 3095613086822, 17571039730136, 99868193737306, 568303494617472
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Comments

Also main diagonal of array: A(i,1)=i, A(1,j)=j; A(i,j) = 2*A(i,j-1) - A(i-1,j). - Benoit Cloitre, Feb 26 2003

Links

Crossrefs

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055826, A055827, A055828, A055829.

Programs

  • Maple
    T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+1, n), n=0..30); # G. C. Greubel, Jan 22 2020
  • Mathematica
    T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+1, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)
  • Sage
    @CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+1, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055826 a(n) = T(2n+2,n), array T as in A055818.

Original entry on oeis.org

1, 11, 60, 328, 1800, 9924, 54964, 305680, 1706256, 9554620, 53653996, 302038488, 1703995800, 9631951476, 54539233380, 309296779296, 1756495236128, 9987721546860, 56857004161180, 324008331785320, 1848182861702184
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055827, A055828, A055829.

Programs

  • Maple
    T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+2, n), n=0..30); # G. C. Greubel, Jan 22 2020
  • Mathematica
    T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+2, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)
  • Sage
    @CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+2, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055828 a(n) = T(2n+4,n), array T as in A055818.

Original entry on oeis.org

1, 47, 336, 2128, 12848, 75808, 441824, 2557024, 14737312, 84726992, 486388912, 2789840688, 15995087184, 91689974592, 525608606912, 3013422222272, 17280193763392, 99117586397296, 568696489153808, 3263973649089808
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055826, A055827, A055829.

Programs

  • Maple
    T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+4, n), n=0..30); # G. C. Greubel, Jan 22 2020
  • Mathematica
    T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+4, n], {n,0,25}] (* G. C. Greubel, Jan 22 2020 *)
  • Sage
    @CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+4, n) for n in (0..30)] # G. C. Greubel, Jan 22 2020

A055829 a(n) = T(2n+5,n), array T as in A055818.

Original entry on oeis.org

1, 95, 768, 5216, 33024, 201792, 1208320, 7145792, 41919744, 244590496, 1421823232, 8243669664, 47708339712, 275738420864, 1592186658816, 9187634766976, 52992487665152, 305556178607328, 1761501729738496
Offset: 0

Views

Author

Clark Kimberling, May 28 2000

Keywords

Links

Crossrefs

Cf. A055818, A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055826, A055827, A055828.

Programs

  • Maple
    T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
  fi; end:
seq(T(n+5, n), n=0..30); # G. C. Greubel, Jan 23 2020
  • Mathematica
    T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+5, n], {n,0,30}] (* G. C. Greubel, Jan 23 2020 *)
  • Sage
    @CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+5, n) for n in (0..30)] # G. C. Greubel, Jan 23 2020
Showing 1-7 of 7 results.

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