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 1-4 of 4 results.

A080851 Square array of pyramidal numbers, read by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 10, 10, 1, 6, 14, 20, 15, 1, 7, 18, 30, 35, 21, 1, 8, 22, 40, 55, 56, 28, 1, 9, 26, 50, 75, 91, 84, 36, 1, 10, 30, 60, 95, 126, 140, 120, 45, 1, 11, 34, 70, 115, 161, 196, 204, 165, 55, 1, 12, 38, 80, 135, 196, 252, 288, 285, 220, 66, 1, 13, 42, 90, 155, 231, 308, 372, 405, 385, 286, 78
Offset: 0

Views

Author

Paul Barry, Feb 21 2003

Keywords

Comments

The first row contains the triangular numbers, which are really two-dimensional, but can be regarded as degenerate pyramidal numbers. - N. J. A. Sloane, Aug 28 2015

Examples

			Array begins (n>=0, k>=0):
1,  3,  6, 10,  15,  21,  28,  36,  45,   55, ... A000217
1,  4, 10, 20,  35,  56,  84, 120, 165,  220, ... A000292
1,  5, 14, 30,  55,  91, 140, 204, 285,  385, ... A000330
1,  6, 18, 40,  75, 126, 196, 288, 405,  550, ... A002411
1,  7, 22, 50,  95, 161, 252, 372, 525,  715, ... A002412
1,  8, 26, 60, 115, 196, 308, 456, 645,  880, ... A002413
1,  9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584

Links

Crossrefs

Numerous sequences in the database are to be found in the array. Rows include the pyramidal numbers A000217, A000292, A000330, A002411, A002412, A002413, A002414, A007584, A007585, A007586.
Columns include or are closely related to A017029, A017113, A017017, A017101, A016777, A017305. Diagonals include A006325, A006484, A002417.
Cf. A057145, A027660 (antidiagonal sums).
See A257199 for another version of this array.

Programs

  • Derive
    vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^4,x,11),x,n),n,0,11),k,-1,10) VECTOR(VECTOR(comb(k+2,2)+comb(k+2,3)n, k, 0, 11), n, 0, 11)
  • Maple
    A080851 := proc(n,k)
    binomial(k+3,3)+(n-1)*binomial(k+2,3) ;
end proc:
seq( seq(A080851(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Oct 01 2021
  • Mathematica
    pyramidalFigurative[ ngon_, rank_] := (3 rank^2 + rank^3 (ngon - 2) - rank (ngon - 5))/6; Table[ pyramidalFigurative[n-k-1, k], {n, 4, 15}, {k, n-3}] // Flatten (* Robert G. Wilson v, Sep 15 2015 *)

Formula

T(n, k) = binomial(k+3, 3) + (n-1)*binomial(k+2, 3), corrected Oct 01 2021.
T(n, k) = T(n-1, k) + C(k+2, 3) = T(n-1, k) + k*(k+1)*(k+2)/6.
G.f. for rows: (1 + n*x)/(1-x)^4, n>=-1.
T(n,k) = sum_{j=1..k+1} A057145(n+2,j). - R. J. Mathar, Jul 28 2016

A185508 Third accumulation array, T, of the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 5, 6, 16, 29, 21, 41, 89, 99, 56, 91, 219, 295, 259, 126, 182, 469, 705, 755, 574, 252, 336, 910, 1470, 1765, 1645, 1134, 462, 582, 1638, 2786, 3605, 3780, 3206, 2058, 792, 957, 2778, 4914, 6706, 7595, 7266, 5754, 3498, 1287, 1507, 4488, 8190, 11634, 13916, 14406, 12894, 9690, 5643, 2002, 2288, 6963, 13035, 19110, 23814, 26068, 25284, 21510, 15510, 8723, 3003, 3367, 10439, 19965, 30030, 38640, 44100
Offset: 1

Views

Author

Clark Kimberling, Jan 29 2011

Keywords

Comments

See A144112 (and A185506) for the definition of accumulation array (aa).
Sequence is aa(aa(aa(A000027))).

Examples

			Northwest corner:
   1    5   16   41   91  182
   6   29   89  219  469  910
  21   99  295  705 1470 2786
  56  259  755 1765 3605 6706

Links

Crossrefs

Cf. A000027, A185506, A185507, A185509.
Cf. A000389 (column 1), A257199 (row 1).

Programs

  • Mathematica
    h[n_,k_]:=k(k+1)(k+2)n(n+1)(n+2)*(4n^2+(5k+23)n+4k^2+3k+41)/2880;
TableForm[Table[h[n,k],{n,1,10},{k,1,15}]]
Table[h[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
  • PARI
    {h(n,k) = k*(k+1)*(k+2)*n*(n+1)*(n+2)*(4*n^2+(5*k+23)*n +4*k^2 +3*k + 41)/2880}; for(n=1,10, for(k=1,n, print1(h(k, n-k+1), ", "))) \\ G. C. Greubel, Nov 23 2017

Formula

T(n,k) = F*(4n^2 + (5k+23)n + 4k^2 + 3k+41), where F = k(k+1)(k+2)n(n+1)(n+2)/2880.

A257200 a(n) = n*(n+1)*(n+2)*(n+3)*(n^2+3*n+26)/720.

Original entry on oeis.org

1, 6, 22, 63, 154, 336, 672, 1254, 2211, 3718, 6006, 9373, 14196, 20944, 30192, 42636, 59109, 80598, 108262, 143451, 187726, 242880, 310960, 394290, 495495, 617526, 763686, 937657, 1143528, 1385824, 1669536, 2000152, 2383689, 2826726, 3336438, 3920631, 4587778, 5347056, 6208384, 7182462
Offset: 1

Views

Author

Luciano Ancora, Apr 18 2015

Keywords

Comments

Antidiagonal sums of the array of 4-dimensional solid numbers shown in Table 3 of Sardelis and Valahas paper (see also Example field).
See A257199 (second comment) for the general formula of this type of numbers: the sequence correspond to the case j = 4.
Binomial transform of (1, 5, 11, 14, 11, 5, 1, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

Examples

			Array in Comments begins:
1,  5, 15,  35,  70, 126, 210,  330, ...
1,  6, 20,  50, 105, 196, 336,  540, ...
1,  7, 25,  65, 140, 266, 462,  750, ...
1,  8, 30,  80, 175, 336, 588,  960, ...
1,  9, 35,  95, 210, 406, 714, 1170, ...
1, 10, 40, 110, 245, 476, 840, 1380, ...

Links

Crossrefs

Cf. A257199, A257201.
See A080852 for another version of the array.

Programs

  • Magma
    [n*(n+1)*(n+2)*(n+3)*(n^2+3*n+26)/720: n in [1..40]]; // Vincenzo Librandi, Apr 18 2015
  • Mathematica
    Table[n (n + 1) (n + 2) (n + 3) (n^2 + 3n + 26)/720, {n, 40}]
  • PARI
    first(m)=vector(m,i,i*(i+1)*(i+2)*(i+3)*(i^2+3*i+26)/720) \\ Anders Hellström, Aug 26 2015
  • PARI
    Vec(x*(-1 + x - x^2)/(-1 + x)^7 + O(x^40)) \\ Michel Marcus, Aug 27 2015

Formula

G.f.: x*(-1 + x - x^2)/(-1 + x)^7.

A257201 a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n^2+4*n+37)/5040.

Original entry on oeis.org

1, 7, 29, 92, 246, 582, 1254, 2508, 4719, 8437, 14443, 23816, 38012, 58956, 89148, 131784, 190893, 271491, 379753, 523204, 710930, 953810, 1264770, 1659060, 2154555, 2772081, 3535767, 4473424, 5616952, 7002776, 8672312, 10672464, 13056153, 15882879, 19219317, 23139948, 27727726, 33074782, 39283166, 46465628
Offset: 1

Views

Author

Luciano Ancora, Apr 18 2015

Keywords

Comments

Antidiagonal sums of the array of 5-dimensional solid numbers (see Example field).
See A257199 (second comment) for the general formula of this type of numbers: the sequence correspond to the case j = 5.
The sequence is the binomial transform of (1, 6, 16, 25, 25, 16, 6, 1, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

Examples

			Array in Comments begins:
  1,  6, 21,  56, 126,  252,  462,  792, 1287, 2002, ...
  1,  7, 27,  77, 182,  378,  714, 1254, 2079, 3289, ...
  1,  8, 33,  98, 238,  504,  966, 1716, 2871, 4576, ...
  1,  9, 39, 119, 294,  630, 1218, 2178, 3663, 5863, ...
  1, 10, 45, 140, 350,  756, 1470, 2640, 4455, 7150, ...
  1, 11, 51, 161, 406,  882, 1722, 3102, 5247, 8437, ...
  1, 12, 57, 182, 462, 1008, 1974, 3564, 6039, 9724, ...
  ...

Links

Crossrefs

Cf. A257199, A257200.

Programs

  • Magma
    [n*(n+1)*(n+2)*(n+3)*(n+4)*(n^2+4*n+37)/5040: n in [1..40]]; // Vincenzo Librandi, Apr 18 2015
  • Mathematica
    Table[n (n + 1) (n + 2) (n + 3) (n + 4) (n^2 + 4n + 37)/5040, {n, 40}]

Formula

G.f.: x*(1 - x + x^2)/(1 - x)^8.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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