A099884 -id:A099884 - cp's OEIS frontend

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), ...

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)).

A276586 Square array A(row,col) = Sum_{k=0..row} binomial(row,k)*A002110(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, 30, 36, 44, 55, 210, 240, 276, 320, 375, 2310, 2520, 2760, 3036, 3356, 3731, 30030, 32340, 34860, 37620, 40656, 44012, 47743, 510510, 540540, 572880, 607740, 645360, 686016, 730028, 777771, 9699690, 10210200, 10750740, 11323620, 11931360, 12576720, 13262736, 13992764, 14770535

Offset: 0

Antti Karttunen, Sep 18 2016

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

Transpose: A276587.
Topmost row: A002110, Leftmost column: A136104.
Cf. A007318, A276085.
Cf. also arrays A066117, A276588, A099884, A255483.

primorial[n_] := Product[Prime[k], {k, 1, n}]; A[n_, k_] := Sum[Binomial[n, j]*primorial[k+j], {j, 0, n}]; Table[A[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 22 2017 *)

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(k, n - k),", ");); print();) \\ Indranil Ghosh, Apr 11 2017

(define (A276586 n) (A276586bi (A002262 n) (A025581 n)))
(define (A276586bi row col) (A276085 (A066117bi (+ 1 row) (+ 1 col))))

A(row,col) = Sum_{k=0..row} binomial(row,k)*A002110(col+k).

A(row,col) = A276085(A066117(row+1,col+1)).

A099896 A permutation of the natural numbers where a(n) = n XOR [n/2] XOR [n/4].

Original entry on oeis.org

1, 3, 2, 7, 6, 4, 5, 14, 15, 13, 12, 9, 8, 10, 11, 28, 29, 31, 30, 27, 26, 24, 25, 18, 19, 17, 16, 21, 20, 22, 23, 56, 57, 59, 58, 63, 62, 60, 61, 54, 55, 53, 52, 49, 48, 50, 51, 36, 37, 39, 38, 35, 34, 32, 33, 42, 43, 41, 40, 45, 44, 46, 47, 112, 113, 115, 114, 119, 118, 116

Offset: 1

Paul D. Hanna, Nov 09 2004

Related to Gray code numbers (A003188) since A003188(n) = n XOR [n/2].

Inverse: A100280; A100281(n) = a(a(n)). - Reinhard Zumkeller, Nov 11 2004

Cf. A003188, A099884.

Array[BitXor[BitXor[#,Floor[#/2]],Floor[#/4]]&,70] (* Harvey P. Dale, Aug 10 2012 *)

PARI
```
a(n)=bitxor(bitxor(n,n\2),n\4)
```

XOR BINOMIAL transform begins: B={1, 2, 3, 7, 7, 0, 0, 14, 14, 0, 0, ...} and continues with B(2^k)=B(2^k+1)=7*2^(k-2) (for k>=2) and zeros elsewhere.

A099902 Multiplies by 2 and shifts right under the XOR BINOMIAL transform (A099901).

Original entry on oeis.org

1, 3, 7, 11, 23, 59, 103, 139, 279, 827, 1895, 2955, 5655, 14395, 24679, 32907, 65815, 197435, 460647, 723851, 1512983, 3881019, 6774887, 9142411, 18219287, 54002491, 123733863, 192940939, 369104407, 939538491, 1610637415, 2147516555

Offset: 0

Paul D. Hanna, Oct 30 2004

Equals the XOR BINOMIAL transform of A099901. Also, equals the main diagonal of the XOR difference triangle A099900, in which the central terms of the rows form the powers of 2.

Bisection of A101624. - Paul Barry, May 10 2005

Robert Israel, Table of n, a(n) for n = 0..3290

Cf. A099884, A099900, A099901.

a:= n -> add((binomial(n-k+floor(k/2),floor(k/2)) mod 2)*2^k, k=0..n):
map(a, [$0..100]); # Robert Israel, Jan 24 2016

{a(n)=local(B);B=0;for(k=0,n,B=bitxor(B,binomial(n-k+k\2,k\2)%2*2^k));B}

a(n)=sum(k=0,n,binomial(n-k+k\2,k\2)%2*2^k)

def A099902(n): return sum(int(not ~((n<<1)-k)&k)<Chai Wah Wu, Jul 30 2025

a(n) = SumXOR_{k=0..n} (binomial(n-k+floor(k/2), floor(k/2)) mod 2)*2^k for n >= 0.
a(n) = SumXOR_{i=0..n} (C(n, i) mod 2)*A099901(n-i), where SumXOR is the analog of summation under the binary XOR operation and C(i, j) mod 2 = A047999(i, j).
a(n) = Sum_{k=0..n} A047999(n-k+floor(k/2), floor(k/2)) * 2^k.
From Paul Barry, May 10 2005: (Start)
a(n) = Sum_{k=0..2n} (binomial(k, 2n-k) mod 2)*2^(2n-k);
a(n) = Sum_{k=0..n} (binomial(2n-k, k) mod 2)*2^k. (End)
a(n) = Sum_{k=0..2n} A106344(2n,k)*2^(2n-k). - Philippe Deléham, Dec 18 2008

A101122 XOR BINOMIAL transform of A101119.

Original entry on oeis.org

7, 17, 0, 34, 0, 0, 0, 68, 0, 0, 0, 0, 0, 0, 0, 159, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 257, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 514, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Simon Plouffe and Paul D. Hanna, Dec 02 2004

nonn

Nonzero terms form A101121 and occur at positions 2^k for k >= 0. A101119 equals the nonzero differences of A006519 and A003484. See A099884 for the definition of the XOR BINOMIAL transform.

Cf. A003484, A006519, A101119, A101120, A101121.

{a(n)=local(B);B=0;for(i=0,n-1,B=bitxor(B,binomial(n-1,i)%2* (16*2^valuation(n-i,2)-2^(valuation(n-i,2)%4)-8*(valuation(n-i,2)\4)-8)));B}

from operator import xor
from functools import reduce
def A101122(n): return reduce(xor,(((1<<(m:=(~(k+1)&k).bit_length()+4))-((m&-4)<<1)-(1<<(m&3)))&-int(not k&~(n-1)) for k in range(n))) # Chai Wah Wu, Jul 10 2022

a(n) = SumXOR_{k=0..n} (C(n, k) mod 2)*A101119(k), where SumXOR is summation under XOR. A101119(n) = SumXOR_{k=0..n} (C(n, k) mod 2)*a(k). a(2^(n-1)) = A101121(n) for n >= 1 and a(k)=0 when k is not a power of 2.

A117998 Decimal number generated by the binary bits of the n-th generation of the Rule 102 elementary cellular automaton.

Original entry on oeis.org

1, 6, 20, 120, 272, 1632, 5440, 32640, 65792, 394752, 1315840, 7895040, 17895424, 107372544, 357908480, 2147450880, 4295032832, 25770196992, 85900656640, 515403939840, 1168248930304, 7009493581824, 23364978606080

Offset: 0

Eric W. Weisstein, Apr 08 2006

Central diagonal of A099884 when viewed as a square array. Thus also a subsequence of A118666. - Antti Karttunen, Jan 18 2020

			1; 1, 1, 0; 1, 0, 1, 0, 0; 1, 1, 1, 1, 0, 0, 0; 1, 0, 0, 0, 1, 0, 0, 0, 0; ...

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Rule 102
Index entries for sequences related to binary expansion of n
Index entries for sequences related to cellular automata
Index entries for sequences related to polynomials in ring GF(2)[X]

Iterates of A048726, starting from a(0) = 1.
Central diagonal of A099884. Bisection of A099885. Subsequence of A118666.
Cf. A000079, A001317, A193231.

NestList[BitXor[4#,2#]&,1,50] (* Paolo Xausa, Oct 04 2023 *)

A117998(n) = (subst(lift(Mod(1+'x, 2)^n), 'x, 2)<Antti Karttunen, Jan 19 2020, after Gheorghe Coserea's code for A001317.

def A117998(n): return sum((bool(~n&n-k)^1)<Chai Wah Wu, May 03 2023

It appears that a(n) = A099885(2*n). - Peter Bala, Feb 01 2017
From Antti Karttunen, Jan 19 2020: (Start)
Bala's observation is correct, and follows from the formula given below and from the fact that this is the central diagonal of square array A099884.
a(n) = A000079(n) * A001317(n). [See Eric Weisstein's World of Mathematics -link]
a(0) = 1; for n > 0, a(n) = A048726(a(n-1)).
(End)

A099889 XOR difference triangle of the odd numbers, read by rows.

Original entry on oeis.org

1, 3, 2, 5, 6, 4, 7, 2, 4, 0, 9, 14, 12, 8, 8, 11, 2, 12, 0, 8, 0, 13, 6, 4, 8, 8, 0, 0, 15, 2, 4, 0, 8, 0, 0, 0, 17, 30, 28, 24, 24, 16, 16, 16, 16, 19, 2, 28, 0, 24, 0, 16, 0, 16, 0, 21, 6, 4, 24, 24, 0, 0, 16, 16, 0, 0, 23, 2, 4, 0, 24, 0, 0, 0, 16, 0, 0, 0, 25, 14, 12, 8, 8, 16, 16, 16, 16, 0

Offset: 0

Paul D. Hanna, Oct 29 2004

Main diagonal is A099890, the XOR BINOMIAL transform of the odd numbers. See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.

			Rows begin:
[1],
[3,2],
[5,6,4],
[7,2,4,0],
[9,14,12,8,8],
[11,2,12,0,8,0],
[13,6,4,8,8,0,0],
[15,2,4,0,8,0,0,0],
[17,30,28,24,24,16,16,16,16],...

Ivan Neretin, Table of n, a(n) for n = 0..8000

Cf. A099884, A099890.

mx = 14; Flatten@Table[NestList[BitXor @@@ Transpose[{Most@#, Rest@#}] &, Range[1, 2 mx, 2], mx][[k, n - k]], {n, 2, mx}, {k, n - 1}] (* Ivan Neretin, Sep 01 2016 *)

T(n,k)=local(B);B=0;for(i=0,k,B=bitxor(B,binomial(k,i)%2*(2*(n-i)+1)));B

T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*(2*(n-i)+1), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i). T(2^n, 2^n) = 2^(n+1) for n>=0, with T(0, 0)=1.

A099894 XOR BINOMIAL transform of A038712.

Original entry on oeis.org

1, 2, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Oct 29 2004

nonn

See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.
a(n) = A062383(n+1) - A062383(n). - Reinhard Zumkeller, Aug 06 2009
A038712 has offset 1, but we need to use offset 0 for the XOR BINOMIAL. - Michael Somos, Dec 30 2016

			G.f. = 1 + 2*x + 4*x^3 + 8*x^7 + 16*x^15 + 32*x^31 + 64*x^63 + 128*x^127 + ...
XOR difference triangle of A038712 begins:
[1],
[3,2],
[1,2,0],
[7,6,4,4],
[1,6,0,4,0],
[3,2,4,4,0,0],
[1,2,0,4,0,0,0],
[15,14,12,12,8,8,8,8],...
where A038712 is in the leftmost column and A099894 (this sequence) forms the main diagonal.
a(1) = 1*1 XOR 0*1 = 1, a(2) = 1*1 XOR 0*3 XOR 1*1 = 0, a(3) = 1*1 XOR 1*3 XOR 1*1 XOR 1*7 = 4 where (1, 3, 1, 7) are the first four terms of A038712. - _Michael Somos_, Dec 30 2016

Cf. A099884, A038712, A099895, A062383, A048298.

a[ n_] := With[ {m = n+1}, If[ m >=0 && Total[ IntegerDigits[ m, 2]] == 1, m, 0]]; (* Michael Somos, Dec 30 2016 *)

{a(n)=local(B);B=0;for(i=0,n,B=bitxor(B,binomial(n,i)%2*A038712(n-i) ));B}

{a(n) = my(m = n+1); m * ( m>=0 && hammingweight(m) == 1)}; /* Michael Somos, Dec 30 2016 */

a(2^n-1) = 2^n for n>=0 and a(k)=0 otherwise. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A038712(n-i) and SumXOR is summation under XOR.

a(n) = A048298(n+1). - Michael Somos, Dec 30 2016

A099900 XOR difference triangle, read by rows, of A099901 (in leftmost column) such that the main diagonal equals A099901 shift left and divided by 2.

Original entry on oeis.org

1, 2, 3, 6, 4, 7, 14, 8, 12, 11, 22, 24, 16, 28, 23, 46, 56, 32, 48, 44, 59, 118, 88, 96, 64, 112, 92, 103, 206, 184, 224, 128, 192, 176, 236, 139, 278, 472, 352, 384, 256, 448, 368, 412, 279, 558, 824, 736, 896, 512, 768, 704, 944, 556, 827, 1654, 1112, 1888, 1408

Offset: 0

Paul D. Hanna, Oct 29 2004

Central terms of rows equal powers of 2: T(n,[n/2]) = 2^n for n>=0. The leftmost column is A099901. The diagonal forms A099902 and equals the XOR BINOMIAL transform of A099901.

			Rows begin:
[_1],
[_2,3],
[6,_4,7],
[14,_8,12,11],
[22,24,_16,28,23],
[46,56,_32,48,44,59],
[118,88,96,_64,112,92,103],
[206,184,224,_128,192,176,236,139],
[278,472,352,384,_256,448,368,412,279],
[558,824,736,896,_512,768,704,944,556,827],
[1654,1112,1888,1408,1536,_1024,1792,1472,1648,1116,1895],...
notice that the column terms equal twice the diagonal (with offset), and that the central terms in the rows form the powers of 2.

Cf. A099884, A099901, A099902.

PARI
```
T(n,k)=if(n
				
```

T(n, [n/2]) = 2^n. T(n+1, 0) = 2*T(n, n) (n>=0); T(0, 0)=1; T(n, k) = T(n, k-1) XOR T(n-1, k-1) for n>k>0. T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*T(n-i, 0), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i).

A277819 Transpose of square array A277820.

Original entry on oeis.org

1, 2, 3, 7, 6, 5, 4, 9, 10, 15, 13, 12, 27, 30, 17, 14, 23, 20, 45, 34, 51, 11, 18, 57, 60, 119, 102, 85, 8, 29, 54, 75, 68, 153, 170, 255, 25, 24, 39, 90, 221, 204, 427, 510, 257, 26, 43, 40, 105, 238, 359, 340, 765, 514, 771, 31, 46, 125, 120, 187, 306, 937, 1020, 1799, 1542, 1285, 28, 33, 114, 135, 136, 461, 854, 1275, 1028, 2313, 2570, 3855

Offset: 1

Antti Karttunen, Nov 01 2016

See A277820.

			The top left 10 x 10 corner of the array:
    1,    2,    7,    4,   13,   14,   11,    8,    25,    26
    3,    6,    9,   12,   23,   18,   29,   24,    43,    46
    5,   10,   27,   20,   57,   54,   39,   40,   125,   114
   15,   30,   45,   60,   75,   90,  105,  120,   135,   150
   17,   34,  119,   68,  221,  238,  187,  136,   393,   442
   51,  102,  153,  204,  359,  306,  461,  408,   667,   718
   85,  170,  427,  340,  937,  854,  599,  680,  1965,  1874
  255,  510,  765, 1020, 1275, 1530, 1785, 2040,  2295,  2550
  257,  514, 1799, 1028, 3341, 3598, 2827, 2056,  6425,  6682
  771, 1542, 2313, 3084, 5911, 4626, 7453, 6168, 11051, 11822

Transpose: A277820.
Row 1: A065621, row 2: A277823.
Column 1: A001317.
Cf. A048724, A099884.

(define (A277819 n) (A277820bi (A004736 n) (A002260 n))) ;; Code for A277820bi given in A277820.

cp's OEIS Frontend

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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A276586 Square array A(row,col) = Sum_{k=0..row} binomial(row,k)*A002110(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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A099896 A permutation of the natural numbers where a(n) = n XOR [n/2] XOR [n/4].

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A099902 Multiplies by 2 and shifts right under the XOR BINOMIAL transform (A099901).

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Maple

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A101122 XOR BINOMIAL transform of A101119.

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A117998 Decimal number generated by the binary bits of the n-th generation of the Rule 102 elementary cellular automaton.

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Mathematica

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A099889 XOR difference triangle of the odd numbers, read by rows.

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