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 1-6 of 6 results.

A080852 Square array of 4D pyramidal numbers, read by antidiagonals.

Original entry on oeis.org

1, 1, 4, 1, 5, 10, 1, 6, 15, 20, 1, 7, 20, 35, 35, 1, 8, 25, 50, 70, 56, 1, 9, 30, 65, 105, 126, 84, 1, 10, 35, 80, 140, 196, 210, 120, 1, 11, 40, 95, 175, 266, 336, 330, 165, 1, 12, 45, 110, 210, 336, 462, 540, 495, 220, 1, 13, 50, 125, 245, 406, 588, 750, 825, 715, 286
Offset: 0

Views

Author

Paul Barry, Feb 21 2003

Keywords

Comments

The first row contains the tetrahedral numbers, which are really three-dimensional, but can be regarded as degenerate 4D pyramidal numbers. - N. J. A. Sloane, Aug 28 2015

Examples

			Array, n >= 0, k >= 0, begins
1 4 10 20  35  56 ...
1 5 15 35  70 126 ...
1 6 20 50 105 196 ...
1 7 25 65 140 266 ...
1 8 30 80 175 336 ...

Links

Crossrefs

Rows include A000292, A000332, A002415, A001296, A002418, A002419, A051740, A051797.
Cf. A057145, A080851, A180266, A055796 (antidiagonal sums).
See A257200 for another version of the array.

Programs

  • Derive
    vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^5,x,11),x,n),n,0,11),k,-1,10)
  • Derive
    VECTOR(VECTOR(comb(k+3,3)+comb(k+3,4)n, k, 0, 11), n, 0, 11)
  • Maple
    A080852 := proc(n,k)
        binomial(k+4,4)+(n-1)*binomial(k+3,4) ;
end proc:
seq( seq(A080852(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Oct 01 2021
  • Mathematica
    T[n_, k_] := Binomial[k+3, 3] + Binomial[k+3, 4]n;
Table[T[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 05 2023 *)

Formula

T(n, k) = binomial(k + 4, 4) + (n-1)*binomial(k + 3, 4), corrected Oct 01 2021.
T(n, k) = T(n - 1, k) + C(k + 3, 4) = T(n - 1, k) + k(k + 1)(k + 2)(k + 3)/24.
G.f. for rows: (1 + nx)/(1 - x)^5, n >= -1.
T(n,k) = sum_{j=0..k} A080851(n,j). - R. J. Mathar, Jul 28 2016

A278807 T(n,k)=Number of nXk 0..2 arrays with rows in nondecreasing lexicographic order and columns in nonincreasing lexicographic order.

Original entry on oeis.org

3, 6, 6, 10, 22, 10, 15, 63, 63, 15, 21, 154, 322, 154, 21, 28, 336, 1439, 1439, 336, 28, 36, 672, 5767, 12958, 5767, 672, 36, 45, 1254, 20972, 110455, 110455, 20972, 1254, 45, 55, 2211, 69834, 870473, 2179956, 870473, 69834, 2211, 55, 66, 3718, 214774, 6275546
Offset: 1

Views

Author

R. H. Hardin, Nov 28 2016

Keywords

Comments

Table starts
..3....6......10.........15............21...............28.................36
..6...22......63........154...........336..............672...............1254
.10...63.....322.......1439..........5767............20972..............69834
.15..154....1439......12958........110455...........870473............6275546
.21..336....5767.....110455.......2179956.........41299256..........725674326
.28..672...20972.....870473......41299256.......1976588468........89730161098
.36.1254...69834....6275546.....725674326......89730161098.....10811999412826
.45.2211..214774...41370842...11698232451....3776527762052...1224101415304069
.55.3718..615120..250517485..172994405326..146582077597322.128795068372302728
.66.6006.1653047.1402100369.2356249442222.5255148833158068

Examples

			Some solutions for n=4 k=4
..1..0..0..0. .1..0..0..0. .1..1..1..0. .1..1..0..0. .1..1..0..0
..1..1..1..0. .1..1..1..0. .2..0..0..2. .2..1..2..2. .2..1..2..2
..1..2..1..0. .1..2..0..1. .2..1..0..0. .2..2..1..1. .2..2..1..0
..2..0..1..2. .1..2..0..1. .2..1..0..2. .2..2..2..2. .2..2..1..1

Links

Crossrefs

Diagonal is A229770.
Column 1 is A000217(n+1).
Column 2 is A257200(n+1).

Formula

Empirical for column k:
k=1: a(n) = (1/2)*n^2 + (3/2)*n + 1
k=2: [polynomial of degree 6]
k=3: [polynomial of degree 16]
k=4: [polynomial of degree 44]

A185509 Fourth accumulation array, T, of the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 6, 7, 22, 41, 28, 63, 146, 161, 84, 154, 406, 561, 476, 210, 336, 966, 1526, 1631, 1176, 462, 672, 2058, 3556, 4361, 3976, 2562, 924, 1254, 4032, 7434, 9996, 10486, 8568, 5082, 1716, 2211, 7392, 14322, 20580, 23716, 22344, 16842, 9372, 3003, 3718, 12837, 25872, 39102, 48216, 49980, 43512, 30822, 16302, 5005, 6006, 21307, 44352, 69762, 90552, 100548, 96432, 79002, 53262, 27027, 8008, 9373, 34034, 72787, 118272, 159852
Offset: 1

Views

Author

Clark Kimberling, Jan 29 2011

Keywords

Comments

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

Examples

			Northwest corner:
1.....6....22....63...154
7....41...146...406...966
28..161...561..1526..3556
84..476..1631..4361..9996

Links

Crossrefs

Cf. A000027, A185506, A185507, A185508.
Cf. A000579 (column 1), A257200 (row 1).

Programs

  • Mathematica
    u[n_,k_]:=k(k+1)(k+2)(k+3)n(n+1)(n+2)(n+3)(5n^2+(6k+39)n+5k^2+9k+86)/86400
TableForm[Table[u[n,k],{n,1,10},{k,1,15}]]
Table[u[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

Formula

T(n,k) = F*(5*n^2 + (6*k + 39)*n + 5*k^2 + 9*k + 86), where
F = k*(k+1)*(k+2)*(k+3)*n*(n+1)*(n+2)*(n+3)/86400.

A257199 a(n) = n*(n+1)*(n+2)*(n^2+2*n+17)/120.

Original entry on oeis.org

1, 5, 16, 41, 91, 182, 336, 582, 957, 1507, 2288, 3367, 4823, 6748, 9248, 12444, 16473, 21489, 27664, 35189, 44275, 55154, 68080, 83330, 101205, 122031, 146160, 173971, 205871, 242296, 283712, 330616, 383537, 443037, 509712, 584193, 667147, 759278, 861328, 974078
Offset: 1

Views

Author

Luciano Ancora, Apr 18 2015

Keywords

Comments

Antidiagonal sums of the array of pyramidal numbers shown in Table 2 of Sardelis and Valahas paper (see A261720).
This is the case j = 3 of (n^2 + (j-1)*n + (j+1)^2 + 1)*binomial(n+j-1, j)/((j+1)*(j+2)), where j is the space dimension: a(n) = (n^2+2*n+17)*binomial(n+2,3)/20.
The sequence is the binomial transform of (1, 4, 7, 7, 4, 1, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

Links

Crossrefs

Cf. A257200, A257201.
For another version of the array, see A080851.

Programs

  • Magma
    [n*(n+1)*(n+2)*(n^2+2*n+17)/120: n in [1..40]]; // Vincenzo Librandi, Apr 18 2015
  • Mathematica
    Table[n (n + 1) (n + 2) (n^2 + 2n + 17)/120, {n, 40}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,5,16,41,91,182},40] (* Harvey P. Dale, Mar 18 2018 *)

Formula

G.f.: x*(1 - x + x^2)/(1 - x)^6.

A261721 Fourth-dimensional figurate numbers.

Original entry on oeis.org

1, 1, 5, 1, 6, 15, 1, 7, 20, 35, 1, 8, 25, 50, 70, 1, 9, 30, 65, 105, 126, 1, 10, 35, 80, 140, 196, 210, 1, 11, 40, 95, 175, 266, 336, 330, 1, 12, 45, 110, 210, 336, 462, 540, 495, 1, 13, 50, 125, 245, 406, 588, 750, 825, 715, 1, 14, 55, 140, 280, 476, 714, 960, 1155, 1210, 1001, 1
Offset: 1

Views

Author

Gary W. Adamson, Aug 30 2015

Keywords

Comments

Generating polygons for the sequences are: Triangle, Square, Pentagon, Hexagon, Heptagon, Octagon, ... .
n-th row sequence is the binomial transform of the fourth row of Pascal's triangle (1,n) followed by zeros; and the fourth partial sum of (1, n, n, n, ...).
n-th row sequence is the binomial transform of:
((n-1) * (0, 1, 3, 3, 1, 0, 0, 0) + (1, 4, 6, 4, 1, 0, 0, 0)).
Given the n-th row of the array (1, b, c, d, ...), the next row of the array is (1, b, c, d, ...) + (0, 1, 5, 15, 35, ...)

Examples

			The array as shown in A257200:
  1,  5, 15,  35,  70, 126, 210,  330, ... A000332
  1,  6, 20,  50, 105, 196, 336,  540, ... A002415
  1,  7, 25,  65, 140, 266, 462,  750, ... A001296
  1,  8, 30,  80, 175, 336, 588,  960, ... A002417
  1,  9, 35,  95, 210, 406, 714, 1170, ... A002418
  1, 10, 40, 110, 245, 476, 840, 1380, ... A002419
  ...
(1, 7, 25, 65, 140, ...) is the third row of the array and is the binomial transform of the fourth row of Pascal's triangle (1,3) followed by zeros: (1, 6, 12, 10, 3, 0, 0, 0, ...); and the fourth partial sum of (1, 3, 3, 3, 0, 0, 0).
(1, 7, 25, 65, 140, ...) is the third row of the array and is the binomial transform of: (2 * (0, 1, 3, 3, 1, 0, 0, 0, ...) + (1, 4, 6, 4, 1, 0, 0, 0, ...)); that is, the binomial transform of (1, 6, 12, 10, 3, 0, 0, 0, ...).
Row 2 of the array is (1, 5, 15, 35, 70, ...) + (0, 1, 5, 15, 35, ...), = (1, 6, 20, 50, 105, ...).

References

  • Albert H. Beiler, "Recreations in the Theory of Numbers"; Dover, 1966, p. 195 (Table 80).

Links

Crossrefs

Cf. A257200, A261720 (pyramidal numbers), A000332, A002415, A001296, A002417, A002418, A002419.
Similar to A080852 but without row n=0.
Main diagonal gives A256859.

Programs

  • Maple
    A:= (n, k)-> binomial(k+3, 3) + n*binomial(k+3, 4):
seq(seq(A(d-k, k), k=0..d-1), d=1..13);  # Alois P. Heinz, Aug 31 2015
  • Mathematica
    row[1] = LinearRecurrence[{5, -10, 10, -5, 1}, {1, 5, 15, 35, 70}, m = 10];
row1 = Join[{0}, row[1] // Most]; row[n_] := row[n] = row[n-1] + row1;
Table[row[n-k+1][[k]], {n, 1, m}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 26 2016 *)
  • PARI
    A(n, k) = binomial(k+3, 3) + n*binomial(k+3, 4)
table(n, k) = for(x=1, n, for(y=0, k-1, print1(A(x, y), ", ")); print(""))
/* Print initial 6 rows and 8 columns as follows: */
table(6, 8) \\ Felix Fröhlich, Jul 28 2016

Formula

G.f. for row n: (1 + (n-1)*x)/(1 - x)^5.
A(n,k) = C(k+3,3) + n * C(k+3,4) = A080852(n,k).
E.g.f. as array: exp(y)*(exp(x)*(24 + 24*(3 + x)*y + 36*(1 + x)*y^2 + 4*(1 + 3*x)*y^3 + x*y^4) - 4*(6 + 18*y + 9*y^2 + y^3))/24. - Stefano Spezia, Aug 15 2024

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 1-6 of 6 results.

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

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