A001803 -id:A001803 - cp's OEIS frontend

A162442 Denominators of the column sums of the EG1 matrix coefficients.

2, 16, 48, 512, 1280, 2048, 14336, 262144, 589824, 2621440, 5767168, 50331648, 109051904, 469762048, 67108864, 34359738368, 73014444032, 103079215104, 652835028992, 1099511627776, 3848290697216, 48378511622144

Offset: 2

Johannes W. Meijer, Jul 06 2009

For the definition of the EG1 matrix coefficients see A162440.

We define the columns sums by cs(n) = sum(EG1[2*m-1,n], m = 1.. infinity) for n => 2.

Equals A052469(n-1)/2 for n=>2.
cs(n) = (1/(n-1))*A001803(n-1)/A046161(n-1) for n=>2.
Cf. A162440 and A162441.

a(n) = denom(cs(n)) and numer(cs(n)) = A162441(n) with cs(n) = (2^(2-2*n)/(n-1))*((2*n-1)!/((n-1)!^2)).

a(n) = denom((1/(n-1))*(2*n-1)*binomial(2*n-2,n-1)/4^(n-1))

A173755 Table read by rows, T(n,k) = (-1)^(n-k)2^(2k-bw(k)), where bw(k) is the binary weight of k (A000120).

Original entry on oeis.org

1, -1, 2, 1, -2, 8, -1, 2, -8, 16, 1, -2, 8, -16, 128, -1, 2, -8, 16, -128, 256, 1, -2, 8, -16, 128, -256, 1024, -1, 2, -8, 16, -128, 256, -1024, 2048, 1, -2, 8, -16, 128, -256, 1024, -2048, 32768, -1, 2, -8, 16, -128, 256, -1024, 2048, -32768, 65536, 1, -2, 8, -16, 128, -256, 1024, -2048, 32768, -65536, 262144

Offset: 0

Paul Curtz, Feb 23 2010

Old name was: Table of the numerators of the higher order differences of the binomial transform of the Madhava-Gregory-Leibniz series for Pi/4.
The binomial transform of 1, -1/3, 1/5, -1/7, 1/9 is given by the sequence A046161(n)/A001803(n).
This sequence of fractions and its higher order differences in the subsequent rows start as:
    1,   2/3,   8/15,   16/35,    128/315,   256/693,   1024/3003,   ...
  -1/3, -2/15, -8/105, -16/315,  -128/3465, -256/9009, -1024/45045,  ...
   1/5,  2/35,  8/315,  16/1155,  128/15015, 256/45045, 1024/255255, ...
  -1/7, -2/64, -8/693, -16/3003, -128/45045, ...
The numerators of this array, read upwards along antidiagonals, define the current sequence.

			Triangle begins:
   1;
  -1,    2;
   1,   -2,    8;
  -1,    2,   -8,   16;
   1,   -2,    8,  -16,  128;
  -1,    2,   -8,   16, -128,  256;
   1,   -2,    8,  -16,  128, -256, 1024;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001803, A005187, A046161.

A173755:= func< n,k | (-1)^(n-k)*2^(k + Valuation(Factorial(k), 2)) >;
[A173755(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 24 2024

A173755 := proc(n,k)
        local L,i;
        L := [seq((-1)^i/(2*i+1),i=0..n+k)] ;
        L := BINOMIAL(L);
        for i from 1 to n do
                L := DIFF(L) ;
        end do:
        op(1+k,L) ;
        numer(%) ;
end proc: # R. J. Mathar, Sep 22 2011
A173755 := proc(n, k) local w; w := proc(n) option remember;
`if`(n=0,1,2^(padic[ordp](2*n,2))*w(n-1)) end: (-1)^(n-k)*w(k) end:
for n from 0 to 8 do seq(A173755(n,k),k=0..n) od; # Peter Luschny, Nov 16 2012

Table[(-1)^(n - k)*2^(2 k - DigitCount[k, 2, 1]), {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 21 2019 *)

def A173755(n,k):
    A005187 = lambda n: A005187(n//2) + n if n > 0 else 0
    return (-1)^(n-k)*2^A005187(k)
for n in (0..8):
    [A173755(n,k) for k in (0..n)]  # Peter Luschny, Nov 16 2012

T(n,k) = (-1)^(n-k)*denom(binomial(-1/2,k)). - Peter Luschny, Nov 21 2012

Simpler definition by Peter Luschny, Nov 21 2012

A225043 Pascal's triangle with row n reduced modulo n+1.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 1, 4, 1, 1, 5, 4, 4, 5, 1, 1, 6, 1, 6, 1, 6, 1, 1, 7, 5, 3, 3, 5, 7, 1, 1, 8, 1, 2, 7, 2, 1, 8, 1, 1, 9, 6, 4, 6, 6, 4, 6, 9, 1, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 1, 11, 7, 9, 6, 6, 6, 6, 9, 7, 11, 1, 1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1

Offset: 0

Roger L. Bagula, Apr 25 2013

The row sums are: {0, 2, 4, 8, 11, 20, 22, 32, 31, 52, 56, ...}.

Since row n is only defined mod n+1, it would seem better to reduce the row sums mod n+1, which gives A062173. - N. J. A. Sloane, Apr 28 2013

			{0},
{1, 1},
{1, 2, 1},
{1, 3, 3, 1},
{1, 4, 1, 4, 1},
{1, 5, 4, 4, 5, 1},
{1, 6, 1, 6, 1, 6, 1},
{1, 7, 5, 3, 3, 5, 7, 1},
{1, 8, 1, 2, 7, 2, 1, 8, 1},
{1, 9, 6, 4, 6, 6, 4, 6, 9, 1},
{1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1},
{1, 11, 7, 9, 6, 6, 6, 6, 9, 7, 11, 1},
{1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1},...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Eric Weisstein's World of Mathematics, Binomial Distribution
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A062173, A001803, A086117.

a225043 n k = a225043_tabl !! n !! k
a225043_row n = a225043_tabl !! n
a225043_tabl = zipWith (map . flip mod) [1..] a007318_tabl
-- Reinhard Zumkeller, Jun 12 2013

Flatten[Table[Mod[Binomial[m, n], m + 1], {m, 0, 12}, {n, 0, m}]]

T(m,n)=binomial(m,n)%(m+1) \\ Charles R Greathouse IV, Apr 25 2013

T(m,n) = binomial(m, n) mod m+1.

Definition edited by N. J. A. Sloane, Apr 28 2013

A004735 Denominator of average distance traveled by n-dimensional fly.

Original entry on oeis.org

1, 3, 4, 15, 8, 35, 64, 315, 128, 693, 512, 3003, 1024, 6435, 16384, 109395, 32768, 230945, 131072, 969969, 262144, 2028117, 2097152, 16900975, 4194304, 35102025, 16777216, 145422675, 33554432

Offset: 1

N. J. A. Sloane

The average distance is actually d(n) = 2*n!!/(n+1)!! if n is odd, and d(n) = (1*Pi)*4*n!!/(n+1)!! if n is even. So a(n) = denominator(d(n)) if n is odd and a(n) = denominator(Pi*d(n)) if n is even. - Michel Marcus, May 24 2013

S. Janson, On the traveling fly problem, Graph Theory Notes of New York, Vol. XXXI, 17, 1996.

S. Janson, On the traveling fly problem.

Cf. A004734.

a(n) = {if (n % 2, eo = 2, eo = 4); denominator(eo*prod(i=0, floor((n-1)/2), n-2*i)/prod(i=0, floor(n/2), n+1-2*i));} \\ Michel Marcus, May 24 2013

a(2n) = A001803(n) (conjectured). - Ralf Stephan, Mar 10 2004

A077595 Numerator of integral from 0 to 1 of (1 + x^2)^n, in lowest terms.

Original entry on oeis.org

1, 4, 28, 96, 1328, 4672, 33472, 121856, 3597056, 13417472, 33655808, 127508480, 5829259264, 22308732928, 171393728512, 660468137984, 40831182635008, 22589996269568, 175323994652672, 681560447647744, 10614717931323392, 289707123275726848, 2261982330593738752

Offset: 0

Michael Somos, Nov 06 2002

nonn

			For n=3 the integral is 96/35, so a(3) = 96.

Cf. A076729.

a[n_] := Numerator[Integrate[(1 + x x)^n, {x, 0, 1}]]
a[n_] := Hypergeometric2F1[-n, 1/2, 3/2, -1]
Table[Numerator[a[n]], {n, 0, 20}] (* Gerry Martens, Aug 09 2015 *)

{a(n) = if( n<0, 0, numerator( subst( intformal((1 + x^2)^n), x, 1)))}

From Fabian Pereyra, Aug 16 2024: (Start)
a(n) = numerator(Sum_{k=0..n} binomial(n,k)/(2*k+1)).
E.g.f.: Sum_{x>=0} a(n)/A001803(n)*x^n/n! = Integral_{z=0..1} e^(x*(1+z^2)) dz. (End)

A086228 Determinant of n X n matrix M(i,j)=binomial(2i+1, j).

Original entry on oeis.org

1, 3, 15, 140, 2520, 88704, 6150144, 843448320, 229417943040, 123987652771840, 133311524260282368, 285432092670742757376, 1217843595395169098137600, 10360289146303272377017958400, 175805226564926843718814452940800

Offset: 0

Benoit Cloitre, Aug 28 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..80

Cf. A001803.

[2^(n*(n-3)/2)*(n+1)*Binomial(2*n+1, n): n in [0..30]]; // G. C. Greubel, Jan 25 2018

Table[2^(n*(n-3)/2)*(n+1)*Binomial[2*n+1, n], {n,0,30}] (* G. C. Greubel, Jan 25 2018 *)

for(n=0, 30, print1(2^(n*(n-3)/2)*(n+1)*binomial(2*n+1, n), ", ")) \\ G. C. Greubel, Jan 25 2018

a(n) = 2^(n*(n-3)/2)*(n+1)*binomial(2*n+1, n)

A162441 Numerators of the column sums of the EG1 matrix coefficients.

Original entry on oeis.org

3, 15, 35, 315, 693, 1001, 6435, 109395, 230945, 969969, 2028117, 16900975, 35102025, 145422675, 20036013, 9917826435, 20419054425, 27981667175, 172308161025, 282585384081, 964378691705, 11835556670925, 24185702762325

Offset: 2

Johannes W. Meijer, Jul 06 2009

For the definition of the EG1 matrix coefficients see A162440.
We define the columns sums by cs(n) = sum(EG1[2*m-1,n], m = 1.. infinity) for n => 2.
The row sums of the EG1 matrix follow the same pattern as those of its even counterpart the EG2 matrix, see A161739 and the formulas.

Equals (2*n-1)*A052468(n-1)
Cf. A162440 and A162442 [denom(cs(n))].
Cf. A161739 (RSEG2 triangle), A001803 and A046161.

a(n) = numer(cs(n)) and denom(cs(n)) = A162442(n) with cs(n) = (2^(2-2*n)/(n-1))*((2*n-1)!/((n-1)!^2)).
cs(n) = 2*EG1[ -1,n]/(n-1) with EG1[ -1,n] = 2^(1-2*n)*(2*n-1)!/((n-1)!^2).
cs(n) = (1/(n-1))*A001803(n-1)/A046161(n-1) for n=>2.
rs(2*m-1,p=0) = sum((n^p)*EG1(2*m-1,n), n = 1..infinity) = 2*zeta(2*m-2) for m =>2.

A173296 Numerators of the inverse binomial transform of the Leibniz series for Pi/4.

Original entry on oeis.org

1, -4, 28, -96, 1328, -4672, 33472, -121856, 3597056, -13417472, 33655808, -127508480, 5829259264, -22308732928, 171393728512, -660468137984, 40831182635008, -22589996269568, 175323994652672, -681560447647744

Offset: 0

Paul Curtz, Feb 15 2010

The series terms for Pi/4 are 1, -1/3, 1/5, -1/7, 1/9, -1/11, + ...

Its inverse binomial transform is 1, -4/3, 28/15, -96/35, 1328/315, -4672/693, + ...

Wikipedia, Leibniz series

Cf. A077595, A173294, A001803.

L := [seq((-1)^n/(2*n+1),n=0..20)] ;
read("transforms") ; BINOMIALi(L) ;
apply(numer,%) ; # R. J. Mathar, Jul 06 2011

a(3) replaced with reduced numerator and a(5) onwards added by R. J. Mathar, Jul 06 2011

A187791 Repeat n+1 times 2^A005187(n).

Original entry on oeis.org

1, 2, 2, 8, 8, 8, 16, 16, 16, 16, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256, 256, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 32768, 32768, 32768, 32768, 32768, 32768, 32768, 32768, 32768, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536

Offset: 0

Paul Curtz, Jan 06 2013

a(n) is the denominators of the antidiagonals of the Lorentz factor, which can be written A001790(n)/A046161(n), and its differences.
1,           1/2,     3/8,      5/16,    35/128,     63/256,...  the Lorentz gamma factor,
-1/2,       -1/8,   -1/16,    -5/128,    -7/256,   -21/1024, ...  -A098597(n)/A046161(n+1),from the Lorentz (beta) factor,
3/8,        1/16,   3/128,     3/256,    7/1024,     9/2048,...  A161200(n+2)/A046161(n+2),
-5/16,    -5/128,  -3/256,   -5/1024,   -5/2048,  -45/32768,...  A161202(n+3)/A046161(n+4),
35/128,    7/256,  7/1024,    5/2048,  35/32768,   35/65536, ...
-63/256, -21/1024, -9/2048, -45/32768, -35/65536, -63/262144, ...  .
Like 1/n and A164555(n)/A027642(n), the Lorentz factor is an autosequence of the second kind. The first column is the signed sequence.
The main diagonal is (-1)^n *A001790(n)/A061549(n).
The Lorentz factor is the differences of (0, followed by A001803(n)) / (1, followed by A046161(n)).
PiSK(n-2)=(0, 0, followed by A001803(n)) / (1, 1, followed by A046161(n)) is also an autosequence of second kind.
Remember that an autosequence of the second kind is a sequence whose inverse binomial transform is the sequence signed, with its main diagonal being the double of its first upper diagonal. - Paul Curtz, Oct 13 2013

			1,
2,   2,
8,   8,  8,
16, 16, 16, 16.

OEIS Wiki, Autosequence
Wikipedia, Lorentz Factor.

Cf. A003506.

Flatten[Table[Denominator[Binomial[2n, n]/4^n], {n, 0, 19}, {n + 1}]] (* Alonso del Arte, Jan 07 2013 *)
(* Checking with the antidiagonals *) diff = Table[ Differences[ CoefficientList[ Series[1/Sqrt[1 - x], {x, 0, 9}], x], n], {n, 0, 9}]; Table[ diff[[n-k+1,k]] // Denominator,{n,0,10},{k,1,n}] // Flatten (* Jean-François Alcover, Jan 07 2013 *)
Flatten[Table[2^IntegerExponent[(2*n)!, 2], {n, 0, 19}, {n + 1}]]; (* Jean-François Alcover, Mar 27 2013, after A005187 *)

Repeat A046161(n) n+1 times. Triangle.

New definition by M. F. Hasler

A269949 Triangle read by rows, T(n,k) = denominator(binomial(-1/2, n-k))*binomial(n-1/2, k-1/2), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 5, 15, 5, 1, 35, 35, 35, 7, 1, 63, 315, 105, 63, 9, 1, 231, 693, 1155, 231, 99, 11, 1, 429, 3003, 3003, 3003, 429, 143, 13, 1, 6435, 6435, 15015, 9009, 6435, 715, 195, 15, 1, 12155, 109395, 36465, 51051, 21879, 12155, 1105, 255, 17, 1

Offset: 0

Peter Luschny, Apr 07 2016

Numerators of "gravitational descendent fields" presented on p. 28 of the Zhou reference. See also p. 31. - Tom Copeland, Feb 13 2017

			Triangle starts:
[  1]
[  1,   1]
[  3,   3,    1]
[  5,  15,    5,   1]
[ 35,  35,   35,   7,  1]
[ 63, 315,  105,  63,  9,  1]
[231, 693, 1155, 231, 99, 11, 1]

J. Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

Cf. A001790 (col. 0), A001803 (col. 1), A161199 (col. 2), A161201 (col. 3).

Cf. A269950.

Table[Denominator[Binomial[-1/2, n - k]] Binomial[n - 1/2, k - 1/2], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 13 2017 *)

A269949 = lambda n,k: binomial(-1/2,n-k).denom()*binomial(n-1/2,k-1/2)
for n in range(8): print([A269949(n,k) for k in (0..n)])

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A162442 Denominators of the column sums of the EG1 matrix coefficients.

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A173755 Table read by rows, T(n,k) = (-1)^(n-k)*2^(2*k-bw(k)), where bw(k) is the binary weight of k (A000120).

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A225043 Pascal's triangle with row n reduced modulo n+1.

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A004735 Denominator of average distance traveled by n-dimensional fly.

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A077595 Numerator of integral from 0 to 1 of (1 + x^2)^n, in lowest terms.

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Mathematica

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A086228 Determinant of n X n matrix M(i,j)=binomial(2i+1, j).

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A162441 Numerators of the column sums of the EG1 matrix coefficients.

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A173296 Numerators of the inverse binomial transform of the Leibniz series for Pi/4.

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A173755 Table read by rows, T(n,k) = (-1)^(n-k)2^(2k-bw(k)), where bw(k) is the binary weight of k (A000120).