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 1-3 of 3 results.

A276953 Square array A(row,col) read by antidiagonals: A(1,col) = A273670(col-1), and for row > 1, A(row,col) = A153880(A(row-1,col)); Dispersion of factorial base shift A153880 (array transposed).

Original entry on oeis.org

1, 3, 2, 4, 8, 6, 5, 12, 30, 24, 7, 14, 48, 144, 120, 9, 26, 54, 240, 840, 720, 10, 32, 126, 264, 1440, 5760, 5040, 11, 36, 150, 744, 1560, 10080, 45360, 40320, 13, 38, 168, 864, 5160, 10800, 80640, 403200, 362880, 15, 50, 174, 960, 5880, 41040, 85680, 725760, 3991680, 3628800, 16, 56, 246, 984, 6480, 46080, 367920, 766080, 7257600, 43545600, 39916800
Offset: 1

Views

Author

Antti Karttunen, Sep 22 2016

Keywords

Comments

The array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
Entries on row n are all multiples of n!. Dividing that factor out gives another array A276616.

Examples

			The top left corner of the array:
    1,    3,     4,     5,     7,     9,    10,    11,    13,    15,    16
    2,    8,    12,    14,    26,    32,    36,    38,    50,    56,    60
    6,   30,    48,    54,   126,   150,   168,   174,   246,   270,   288
   24,  144,   240,   264,   744,   864,   960,   984,  1464,  1584,  1680
  120,  840,  1440,  1560,  5160,  5880,  6480,  6600, 10200, 10920, 11520
  720, 5760, 10080, 10800, 41040, 46080, 50400, 51120, 81360, 86400, 90720

Links

Crossrefs

Inverse permutation: A276954.
Transpose: A276955.
Cf. A276949 (index of row where n appears), A276951 (index of column).
Cf. A153880, A273670.
Row 1: A273670, Row 2: A276932, Row 3: A276933.
Column 1: A000142. For other columns, see the rows of transposed array A276955.
Related or similar permutations: A257503, A275848, A273666.
Cf. also arrays A276616, A276589 & A276943.

Programs

Formula

A(1,col) = A273670(col-1), and for row > 1, A(row,col) = A153880(A(row-1,col))
As a composition of other permutations:
a(n) = A275848(A257503(n)).
Other identities. For all n >= 1:
A(A276949(n),A276951(n)) = n.

A276588 Square array A(row,col) = Sum_{k=0..row} binomial(row,k)*(1+col+k)!, read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

1, 2, 3, 6, 8, 11, 24, 30, 38, 49, 120, 144, 174, 212, 261, 720, 840, 984, 1158, 1370, 1631, 5040, 5760, 6600, 7584, 8742, 10112, 11743, 40320, 45360, 51120, 57720, 65304, 74046, 84158, 95901, 362880, 403200, 448560, 499680, 557400, 622704, 696750, 780908, 876809, 3628800, 3991680, 4394880, 4843440, 5343120, 5900520, 6523224, 7219974, 8000882, 8877691
Offset: 0

Views

Author

Antti Karttunen, Sep 19 2016

Keywords

Examples

			The top left corner of the array:
     1,     2,     6,     24,     120,      720,      5040,      40320
     3,     8,    30,    144,     840,     5760,     45360,     403200
    11,    38,   174,    984,    6600,    51120,    448560,    4394880
    49,   212,  1158,   7584,   57720,   499680,   4843440,   51932160
   261,  1370,  8742,  65304,  557400,  5343120,  56775600,  661933440
  1631, 10112, 74046, 622704, 5900520, 62118720, 718709040, 9059339520

Links

Crossrefs

Transpose: A276589.
Topmost row (row 0): A000142, Row 1: A001048 (without its initial 2), Row 2: A001344 (from a(1) = 11 onward), Row 3: A001345 (from a(1) = 49 onward), Row 4: A001346 (from a(1) = 261 onward), Row 5: A001347 (from a(1) = 1631 onward).
Leftmost column (column 0): A001339, Column 1: A001340, Columns 2-3: A001341 & A001342 (apparently).
Cf. A276075.
Cf. also arrays A066117, A276586, A099884, A255483.

Programs

  • Mathematica
    T[r_, c_]:=Sum[Binomial[r, k](1 + c + k)!, {k, 0, r}]; Table[T[c, r - c], {r, 0, 10}, {c, 0, r}] // Flatten (* Indranil Ghosh, Apr 11 2017 *)
  • PARI
    T(r, c) = sum(k=0, r, binomial(r, k)*(1 + c + k)!);
for(r=0, 10, for(c=0, r, print1(T(c, r - c),", ");); print();) \\ Indranil Ghosh, Apr 11 2017
  • Python
    from sympy import binomial, factorial
def T(r, c): return sum([binomial(r, k) * factorial(1 + c + k) for k in range(r + 1)])
for r in range(11): print([T(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Apr 11 2017
  • Scheme
    (define (A276588 n) (A276588bi (A002262 n) (A025581 n)))
(define (A276588bi row col) (A276075 (A066117bi (+ 1 row) (+ 1 col)))) ;; Code for A066117bi given in A066117, and for A276075 under the respective entry.

Formula

A(row,col) = Sum_{k=0..row} binomial(row,k)*A000142(1+col+k).
A(row,col) = A276075(A066117(row+1,col+1)).

A276587 Transpose of square array A276586.

Original entry on oeis.org

1, 3, 2, 11, 8, 6, 55, 44, 36, 30, 375, 320, 276, 240, 210, 3731, 3356, 3036, 2760, 2520, 2310, 47743, 44012, 40656, 37620, 34860, 32340, 30030, 777771, 730028, 686016, 645360, 607740, 572880, 540540, 510510, 14770535, 13992764, 13262736, 12576720, 11931360, 11323620, 10750740, 10210200, 9699690
Offset: 0

Views

Author

Antti Karttunen, Sep 18 2016

Keywords

Comments

Rows give the successive first differences of A136104.

Examples

			The top left corner of the array:
      1,      3,       11,        55,        375,         3731
      2,      8,       44,       320,       3356,        44012
      6,     36,      276,      3036,      40656,       686016
     30,    240,     2760,     37620,     645360,     12576720
    210,   2520,    34860,    607740,   11931360,    277008480
   2310,  32340,   572880,  11323620,  265077120,   7687412040
  30030, 540540, 10750740, 253753500, 7422334920, 235239464460

Links

Crossrefs

Topmost row: A136104. For other rows and columns, see the information given in transpose A276586.
Cf. also A276589.

Programs

  • PARI
    P(n)=prod(i=1, n, prime(i));
T(n, k) = sum(j=0, n, binomial(n, j)*P(k + j));
for(n=0, 10, for(k=0, n, print1(T(n - k, k),", ");); print();) \\ Indranil Ghosh, Apr 11 2017
  • Scheme
    (define (A276587 n) (A276586bi (A025581 n) (A002262 n))) ;; Code for A276586bi given in A276586.
Showing 1-3 of 3 results.

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