A276586 -id:A276586 - cp's OEIS frontend

A276587 Transpose of square array A276586.

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

Antti Karttunen, Sep 18 2016

Rows give the successive first differences of A136104.

			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

Index entries for sequences related to primorial numbers

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

Cf. also A276589.

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

(define (A276587 n) (A276586bi (A025581 n) (A002262 n))) ;; Code for A276586bi given in A276586.

A099884 XOR difference triangle of the powers of 2, read by rows; Square array A(row,col): A(0,col) = 2^col, A(row,col) = A048724(A(row-1, col)) for row > 0, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 12, 10, 15, 16, 24, 20, 30, 17, 32, 48, 40, 60, 34, 51, 64, 96, 80, 120, 68, 102, 85, 128, 192, 160, 240, 136, 204, 170, 255, 256, 384, 320, 480, 272, 408, 340, 510, 257, 512, 768, 640, 960, 544, 816, 680, 1020, 514, 771, 1024, 1536, 1280, 1920

Offset: 0

Paul D. Hanna, Oct 28 2004

Define an "XOR difference triangle" for a sequence A by the following process. Start with A in the leftmost column. Generate the next column by performing the XOR operation between adjacent terms of the prior column. Repeat this process to generate the XOR difference triangle for A. Further, we define the "XOR BINOMIAL transform" of A as the main diagonal in the XOR difference triangle for A. The XOR BINOMIAL transform is its self-inverse. Let a sequence B be the XOR BINOMIAL transform of A, then we may express B by: B(n) = SumXOR_{k=0..n} A047999(n,k)*A(k), which is equivalent to: B(n) = (C(n,0)mod 2)*A(0) XOR (C(n,1)mod 2)*A(1) XOR (C(n,2)mod 2)*A(2) XOR ... XOR (X(n,n)mod 2)*A(n), where the coefficients are C(n,k)(mod 2) = A047999(n,k).

This sequence is a rearrangement of the numbers which are 2^k times distinct Fermat numbers (numbers of the form 2^(2^m) + 1). This matches the sizes of polygons constructible with compass and straightedge (A003401) up to 2^32+1, which is the first nonprime Fermat number. - Franklin T. Adams-Watters, Jun 16 2006

			The main diagonal equals A001317 (Pascal's triangle mod 2 in decimal):
{1,3,5,15,17,51,85,255,257,771,1285,3855,...}, and defines the XOR BINOMIAL transform of the powers of 2.
Rows begin:
  1;
  2, 3;
  4, 6, 5;
  8, 12, 10, 15;
  16, 24, 20, 30, 17;
  32, 48, 40, 60, 34, 51;
  64, 96, 80, 120, 68, 102, 85;
  128, 192, 160, 240, 136, 204, 170, 255;
  256, 384, 320, 480, 272, 408, 340, 510, 257;
  512, 768, 640, 960, 544, 816, 680, 1020, 514, 771;
  1024, 1536, 1280, 1920, 1088, 1632, 1360, 2040, 1028, 1542, 1285;
  2048, 3072, 2560, 3840, 2176, 3264, 2720, 4080, 2056, 3084, 2570, 3855;
  ...
From _Antti Karttunen_, Sep 19 2016: (Start)
Viewed as a square array, the top left corner looks like this:
     1,    2,     4,     8,    16,     32,     64,    128
     3,    6,    12,    24,    48,     96,    192,    384
     5,   10,    20,    40,    80,    160,    320,    640
    15,   30,    60,   120,   240,    480,    960,   1920
    17,   34,    68,   136,   272,    544,   1088,   2176
    51,  102,   204,   408,   816,   1632,   3264,   6528
    85,  170,   340,   680,  1360,   2720,   5440,  10880
   255,  510,  1020,  2040,  4080,   8160,  16320,  32640
   257,  514,  1028,  2056,  4112,   8224,  16448,  32896
   771, 1542,  3084,  6168, 12336,  24672,  49344,  98688
  1285, 2570,  5140, 10280, 20560,  41120,  82240, 164480
  3855, 7710, 15420, 30840, 61680, 123360, 246720, 493440
  4369, 8738, 17476, 34952, 69904, 139808, 279616, 559232
  ...
(End)
The square array shown above can be viewed as a subtable of a multiplication table with particular relevance to the carryless multiplication defined by A048720, as the first column gives the A048720 powers of 3 (and the first row gives powers of 2, which are the same as in standard arithmetic). - _Peter Munn_, Jan 13 2020

Paul D. Hanna, First 45 Rows of Triangle, in flattened form.
SeqFan-mailing list, Discussion of about related array A255483
Index entries for sequences related to polynomials in ring GF(2)[X]

Essentially GF(2)[X] analog of table A036561. - Antti Karttunen, Jan 18 2020
Cf. A047999, A158875 (row sums).
Cf. A000215, A003401, A048675, A048720, A048724, A066117, A248663, A255483, A276586.
Cf. A000079 (first column of triangular table, the topmost row of square array).
Cf. A001317 (the rightmost diagonal of triangular table, the leftmost column of square array).
Cf. A099885, A117998 (central diagonals).
Cf. A276618 (transpose), A091202, A193231.

a[n_]:= Sum[Mod[Binomial[n, i], 2]*2^i, {i, 0, n}]; T[n_, k_]:=2^(n - k)a[k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, Apr 11 2017 *)

{T(n,k)=local(B);B=0;for(i=0,k,B=bitxor(B,binomial(k,i)%2*2^(n-i)));B}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

from sympy import binomial
def a(n):
    return sum((binomial(n, i)%2)*2**i for i in range(n + 1))
def T(n, k): return 2**(n - k)*a(k)
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 11 2017

(define (A099884 n) (A099884bi (A002262 n) (A025581 n)))
;; Then use either this recurrence:
(define (A099884bi row col) (if (zero? row) (A000079 col) (A048724 (A099884bi (- row 1) col))))
;; or this one:
(define (A099884bi row col) (if (zero? col) (A001317 row) (* 2 (A099884bi row (- col 1)))))
;; Antti Karttunen, Sep 19 2016

T(n, k) = 2^(n-k)*A001317(k). T(n, n) = A001317(n) = SumXOR_{k=0..n} A047999(n, k)*2^k, where SumXOR is the analog of summation under the binary XOR operation.
From Antti Karttunen, Sep 19 2016: (Start)
When viewed as a square array A(row,col), with row >= 0, col >= 0, the following recurrences and formulas are valid:
A(0,col) = A000079(col), for row > 0, A(row,col) = A048724(A(row-1, col)).
A(row,0) = A001317(row), for col > 0, A(row,col) = 2*A(row,col-1).
A(row,col) = A248663(A066117(row+1,col+1)) = A048675(A255483(row,col+1)).
(End)
With the definitions from Antti Karttunen above, A(row+1, col) = A048720(3, A(row, col)). - Peter Munn, Jan 13 2020
A(n,k) = A193231(A(k,n)) = A091202(A036561(n,k)). - Antti Karttunen, Jan 18 2020

Square array interpretation added as a second, alternative description by Antti Karttunen, Sep 19 2016

A066117 Triangle read by rows: T(n,k) = T(n-1,k-1)*T(n,k-1) and T(n,1) = prime(n).

Original entry on oeis.org

2, 3, 6, 5, 15, 90, 7, 35, 525, 47250, 11, 77, 2695, 1414875, 66852843750, 13, 143, 11011, 29674645, 41985913344375, 2806877704512541816406250, 17, 221, 31603, 347980633, 10326201751150285, 433555011900329243987584396875

Offset: 1

Henry Bottomley, Dec 05 2001

As a square array read by descending antidiagonals, A(n, k), n >= 1, k >= 1, gives the encoding defined in A297845 of the polynomial (x+1)^(n-1) * x^(k-1). - Peter Munn, Jul 27 2022

			T(4,3) = T(3,2)*T(4,2) = 15*35 = 525. Rows start
     2;
    3, 6;
  5, 15, 90;
7, 35, 525, 47250;
...
From _Antti Karttunen_, Sep 18 2016: (Start)
Alternatively, this table can be viewed as a square array. Then the top left 5x4 corner looks as:
    2,       3,        5,         7,         11
    6,      15,       35,        77,        143
   90,     525,     2695,     11011,      31603
47250, 1414875, 29674645, 347980633, 2255916949
(End)

Index entries for triangles and arrays related to Pascal's triangle

Cf. A000040, A006094 and A066116 (three leftmost diagonal of triangular table = three topmost rows of square array).
Cf. A007188, A267096 (two rightmost diagonals of the triangular table = two leftmost columns of square array).
Cf. A003961, A055396, A297845.
Cf. A064319, A066119.
Cf. also A099884, A255483, A276586, A276588 (other arrays derived from this one).

T[n_, 1] := Prime[n];
T[n_, k_] := T[n, k] = T[n - 1, k - 1]*T[n, k - 1];
Table[T[n, k], {n, 1, 7}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 13 2017 *)

(define (A066117 n) (A066117bi (A002260 n) (A004736 n)))
;; Compute as a square array, with row >= 1, col >= 1:
(define (A066117bi row col) (if (= 1 row) (A000040 col) (* (A066117bi (- row 1) col) (A066117bi (- row 1) (+ col 1)))))
;; With alternative recurrence:
(define (A066117bi row col) (if (= 1 col) (A007188 (- row 1)) (A003961 (A066117bi row (- col 1)))))
;; Antti Karttunen, Sep 18 2016

From Antti Karttunen, Sep 19 2016: (Start)
When computed as a square array A(row,col), row >= 1, col >= 1:
A(1,col) = A000040(col), for row > 1, A(row,col) = A(row-1,col)*A(row-1,col+1).
A(row,1) = A007188(row-1), for col > 1, A(row,col) = A003961(A(row,col-1)).
For all row >= 1, col >= 1, A055396(A(row,col)) = col.
(End)
A(1,1) = 2; for n > 1, A(n,k) = A297845(A(n-1,k),6); for k > 1, A(n,k) = A297845(A(n,k-1),3). - Peter Munn, Jul 20 2022

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

Antti Karttunen, Sep 19 2016

			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

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.

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 *)

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

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

(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.

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

A(row,col) = A276075(A066117(row+1,col+1)).

A136104 A007318 * A002110; a(n) = Sum_{k=0..n} binomial(n,k)*A002110(k).

Original entry on oeis.org

1, 3, 11, 55, 375, 3731, 47743, 777771, 14770535, 331611235, 9205305591, 285781156343, 10308779559631, 418386835375575, 18097509979840775, 846748292083023991, 44182142790019823943, 2570069981187508600331, 157428743473326543397855, 10449715795107936675445215, 739751959772798881608189731

Offset: 0

Gary W. Adamson, Dec 14 2007

nonn

			a(3) = 55 = (1, 3, 3, 1) dot (1, 2, 6, 30) = (1 + 6 + 18 + 30), where A002110 = (1, 2, 6, 30, 210, 2310, ...).

Cf. A002110, A007188, A007318, A276085.
Leftmost column of A276586.
Cf. also A001339, A121572.

b:= proc(n) option remember; `if`(n=0, 1, ithprime(n)*b(n-1)) end:
a:= n-> add(binomial(n, k)*b(k), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 20 2016

b[n_] := b[n] = If[n==0, 1, Prime[n]*b[n-1]]; a[n_] := Sum[Binomial[n, k]*b[k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

Binomial transform of primorial numbers, A002110.

a(n) = A276085(A007188(n)). - Antti Karttunen, Sep 18 2016

A few more terms from L. Edson Jeffery, Apr 11 2011

Explicit binomial sum formula added to the name by Antti Karttunen, Sep 19 2016

cp's OEIS Frontend

A276587 Transpose of square array A276586.

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A099884 XOR difference triangle of the powers of 2, read by rows; Square array A(row,col): A(0,col) = 2^col, A(row,col) = A048724(A(row-1, col)) for row > 0, read by descending antidiagonals.

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A066117 Triangle read by rows: T(n,k) = T(n-1,k-1)*T(n,k-1) and T(n,1) = prime(n).

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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), ...

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A136104 A007318 * A002110; a(n) = Sum_{k=0..n} binomial(n,k)*A002110(k).

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