A002176 -id:A002176 - cp's OEIS frontend

A002177 Numerators of Cotesian numbers (not in lowest terms): A002176(n)*C(n,0).

1, 1, 1, 7, 19, 41, 751, 989, 2857, 16067, 2171465, 1364651, 8181904909, 90241897, 35310023, 15043611773, 55294720874657, 203732352169, 69028763155644023, 19470140241329, 1022779523247467, 396760150748100749

Offset: 1

N. J. A. Sloane

W. W. Johnson, On Cotesian numbers: their history, computation and values to n=20, Quart. J. Pure Appl. Math., 46 (1914), 52-65.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
W. M. Johnson, On Cotesian numbers: their history, computation and values to n=20, Quart. J. Pure Appl. Math., 46 (1914), 52-65. [Annotated scanned copy]

Cf. A100640/A100641, A100620/A100621, A002176-A002179.

cn[n_, 0] := Sum[ n^j*StirlingS1[n, j]/(j+1), {j, 1, n+1}]/n!; cn[n_, n_] := cn[n, 0]; cn[n_, k_] := 1/n!*Binomial[n, k]* Sum[ n^(j+m)*StirlingS1[k, j]* StirlingS1[n-k, m]/((m+1)*Binomial[j+m+1, m+1]), {m, 1, n}, {j, 1, k+1}]; a[n_] := cn[n, 0]*LCM @@ Table[ Denominator[cn[n, k]], {k, 0, n}]; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Oct 25 2011 *)

cn(n) = mattranspose( matinverseimage( matrix(n+1, n+1, k, m, (m-1)^(k-1)), matrix(n+1, 1, k, m, n^(k-1)/k)))[ 1, ]; \\ vector of quadrature formula coefficients via matrix solution

PARI
```
ncn(n) = denominator(cn(n)) * cn(n);
```

nk(n,k) = if(k<0 || k>n, 0, ncn(n)[ k+1 ]);

PARI
```
A002177(n) = nk(n,0);
```

More terms from Michael Somos

A002179 Numerators of Cotesian numbers (not in lowest terms): A002176*C(n,2).

Original entry on oeis.org

0, 1, 3, 12, 50, 27, 1323, -928, 1080, -48525, -3237113, -7587864, -31268252574, -770720657, -232936065, -179731134720, -542023437008852, -3212744374395, -926840515700222955, -389358194177500, -17858352159793110

Offset: 2

N. J. A. Sloane

W. W. Johnson, On Cotesian numbers: their history, computation and values to n=20, Quart. J. Pure Appl. Math., 46 (1914), 52-65.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. M. Johnson, On Cotesian numbers: their history, computation and values to n=20, Quart. J. Pure Appl. Math., 46 (1914), 52-65. [Annotated scanned copy]

Cf. A100640/A100641, A100620/A100621, A002176, A002177, A002178.

cn[n_, 0] := Sum[n^j*StirlingS1[n, j]/(j+1), {j, 1, n+1}]/n!; cn[n_, n_] := cn[n, 0]; cn[n_, k_] := 1/n!*Binomial[n, k]*Sum[n^(j+m)*StirlingS1[k, j]*StirlingS1[n-k, m]/((m+1)*Binomial[j+m+1, m+1]), {m, 1, n}, {j, 1, k+1}]; A002176[n_] := LCM @@ Table[Denominator[cn[n, k]], {k, 0, n}]; a[2] = 0; a[n_] := A002176[n-1]*cn[n-1, 2]; Table[a[n], {n, 2, 22}] (* Jean-François Alcover, Oct 08 2013 *)

cn(n)= mattranspose(matinverseimage( matrix(n+1,n+1,k,m,(m-1)^(k-1)),matrix(n+1,1,k,m,n^(k-1)/k)))[ 1, ] \\ vector of quadrature formula coefficients via matrix solution

ncn(n)= denominator(cn(n))*cn(n); nk(n,k)= if(k<0 || k>n,0,ncn(n)[ k+1 ]); A002177(n)= nk(n,2)

More terms from Michael Somos

A002178 Numerators of Cotesian numbers (not in lowest terms): A002176*C(n,1).

Original entry on oeis.org

1, 4, 3, 32, 75, 216, 3577, 5888, 15741, 106300, 13486539, 9903168, 56280729661, 710986864, 265553865, 127626606592, 450185515446285, 1848730221900, 603652082270808125, 187926090380000, 9545933933230947

Offset: 1

N. J. A. Sloane

W. W. Johnson, On Cotesian numbers: their history, computation and values to n=20, Quart. J. Pure Appl. Math., 46 (1914), 52-65.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. M. Johnson, On Cotesian numbers: their history, computation and values to n=20, Quart. J. Pure Appl. Math., 46 (1914), 52-65. [Annotated scanned copy]

Cf. A100640/A100641, A100620/A100621, A002176-A002179.

cn[n_, 0] := Sum[n^j*StirlingS1[n, j]/(j+1), {j, 1, n+1}]/n!; cn[n_, n_] := cn[n, 0]; cn[n_, k_] := 1/n!*Binomial[n, k]*Sum[n^(j+m)*StirlingS1[k, j]* StirlingS1[n-k, m]/((m+1)*Binomial[j+m+1, m+1]), {m, 1, n}, {j, 1, k+1}]; A002176[n_] := LCM @@ Table[Denominator[cn[n, k]], {k, 0, n}]; a[2] = 0; a[n_] := A002176[n]*cn[n, 1]; Table[a[n], {n, 1, 21}] (* Jean-François Alcover, Oct 08 2013 *)

cn(n)= mattranspose(matinverseimage( matrix(n+1,n+1,k,m,(m-1)^(k-1)),matrix(n+1,1,k,m,n^(k-1)/k)))[ 1, ] \\ vector of quadrature formula coefficients via matrix solution

ncn(n)= denominator(cn(n))*cn(n); nk(n,k)= if(k<0 || k>n,0,ncn(n)[ k+1 ]); A002177(n)= nk(n,1)

More terms from Michael Somos

A100642 Triangle read by rows: numerators of Cotesian numbers C(n,k) (0 <= k <= n) if the denominators are set to the lcm's of the rows (A002176).

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 1, 3, 3, 1, 7, 32, 12, 32, 7, 19, 75, 50, 50, 75, 19, 41, 216, 27, 272, 27, 216, 41, 751, 3577, 1323, 2989, 2989, 1323, 3577, 751, 989, 5888, -928, 10496, -4540, 10496, -928, 5888, 989, 2857, 15741, 1080, 19344, 5778, 5778, 19344, 1080, 15741, 2857, 16067

Offset: 0

N. J. A. Sloane, Dec 04 2004

			0, 1/2, 1/2, 1/6, 2/3, 1/6, 1/8, 3/8, 3/8, 1/8, 7/90, 16/45, 2/15, 16/45, 7/90, 19/288, 25/96, 25/144, 25/144, 25/96, 19/288, 41/840, 9/35, 9/280, 34/105, 9/280, 9/35, 41/840, ... = A100640/A100641 = A100642/A002176 (the latter is not in lowest terms)
Triangle begins
0;
1, 1;
1, 4, 1;
1, 3, 3, 1;
7, 32, 12, 32, 7;

Carl Erik Froeberg, Numerical Mathematics, Benjamin/Cummings Pu.Co. 1985, ISBN 0-8053-2530-1, Chapter 17.2.
Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.

Alois P. Heinz, Rows n = 0..100, flattened
W. M. Johnson, On Cotesian numbers: their history, computation and values to n=20, Quart. J. Pure Appl. Math., 46 (1914), 52-65. [Annotated scanned copy]

Cf. A100641, A100620, A100621, A002177, A002176 (row sums), A100640.

# (This defines the Cotesian numbers C(n,i))
with(combinat); C:=proc(n,i) if i=0 or i=n then RETURN( (1/n!)*add(n^a*stirling1(n,a)/(a+1),a=1..n+1) ); fi; (1/n!)*binomial(n,i)* add( add( n^(a+b)*stirling1(i,a)*stirling1(n-i,b)/((b+1)*binomial(a+b+1,b+1)), b=1..n-i+1), a=1..i+1); end;
den:=proc(n) local t1,i; t1:=1; for i from 0 to n do t1:=ilcm(t1,denom(C(n,i))); od: t1; end;
# Then den(n)*C(n,k) gives the current sequence
seq(seq(den(n,k)*C(n,k), k=0..n), n=0..10);

c[n_, i_] /; i == 0 || i == n = (1/n!)*Sum[n^a*StirlingS1[n, a]/(a+1), {a, 1, n+1}]; c[n_, i_] = (1/n!)*Binomial[n, i]*Sum[n^(a + b)*StirlingS1[i, a]*StirlingS1[n-i, b]/((b+1)*Binomial[a+b+1, b+1]), {b, 1, n}, {a, 1, i+1}]; den[n_] := (For[t1 = 1; i = 0, i <= n, i++, t1 = LCM[t1, c[n, i] // Denominator]]; t1); Table[den[n]*c[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 11 2013, after Maple *)

A100641 Triangle read by rows: denominators of Cotesian numbers C(n,k) (0 <= k <= k).

Original entry on oeis.org

1, 2, 2, 6, 3, 6, 8, 8, 8, 8, 90, 45, 15, 45, 90, 288, 96, 144, 144, 96, 288, 840, 35, 280, 105, 280, 35, 840, 17280, 17280, 640, 17280, 17280, 640, 17280, 17280, 28350, 14175, 14175, 14175, 2835, 14175, 14175, 14175, 28350, 89600, 89600, 2240, 5600, 44800, 44800, 5600

Offset: 0

N. J. A. Sloane, Dec 04 2004

			Triangle A100640/A100641 begins:
[1],
[1/2, 1/2],
[1/6, 2/3, 1/6],
[1/8, 3/8, 3/8, 1/8],
[7/90, 16/45, 2/15, 16/45, 7/90],
[19/288, 25/96, 25/144, 25/144, 25/96, 19/288],
[41/840, 9/35, 9/280, 34/105, 9/280, 9/35, 41/840],
[751/17280, 3577/17280, 49/640, 2989/17280, 2989/17280, 49/640, 3577/17280, 751/17280],
...

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
L. M. Milne-Thompson, Calculus of Finite Differences, MacMillan, 1951, p. 170.

Cf. A100640-A100648, A100620, A100621, A002177, A002176.

(This defines the Cotesian numbers C(n,i)) with(combinat); C:=proc(n,i) if i=0 or i=n then RETURN( (1/n!)*add(n^a*stirling1(n,a)/(a+1),a=1..n+1) ); fi; (1/n!)*binomial(n,i)* add( add( n^(a+b)*stirling1(i,a)*stirling1(n-i,b)/((b+1)*binomial(a+b+1,b+1)), b=1..n-i+1), a=1..i+1); end;
# Another program:
T:=proc(n,k) (-1)^(n-k)*(n/(n-1))*binomial(n-1,k-1)* integrate(expand(binomial(t-1,n))/(t-k), t=1..n); end;
[[1], seq( [seq(T(n,k),k=1..n)], n=2..14)];

a[n_, i_] /; i == 0 || i == n = 1/n!*Sum[n^a StirlingS1[n, a]/(a + 1), {a, 1, n + 1}]; a[n_, i_] = 1/n!*Binomial[n, i] Sum[n^(a + b)*StirlingS1[i, a]*StirlingS1[n - i, b]/((b + 1)*Binomial[a + b + 1, b + 1]), {b, 1, n}, {a, 1, i + 1}]; Table[a[n, i], {n, 0, 10}, {i, 0, n}] // Flatten // Denominator // Take[#, 52] &
(* Jean-François Alcover, May 17 2011, after Maple prog. *)

A100640 Triangle read by rows: numerators of Cotesian numbers C(n,k) (0 <= k <= n).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 3, 3, 1, 7, 16, 2, 16, 7, 19, 25, 25, 25, 25, 19, 41, 9, 9, 34, 9, 9, 41, 751, 3577, 49, 2989, 2989, 49, 3577, 751, 989, 2944, -464, 5248, -454, 5248, -464, 2944, 989, 2857, 15741, 27, 1209, 2889, 2889, 1209, 27, 15741, 2857, 16067, 26575, -16175, 5675

Offset: 0

N. J. A. Sloane, Dec 04 2004

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
L. M. Milne-Thompson, Calculus of Finite Differences, MacMillan, 1951, p. 170.

Cf. A100641-A100648, A100620, A100621, A002177, A002176.

(This defines the Cotesian numbers C(n,i)) with(combinat); C:=proc(n,i) if i=0 or i=n then RETURN( (1/n!)*add(n^a*stirling1(n,a)/(a+1),a=1..n+1) ); fi; (1/n!)*binomial(n,i)* add( add( n^(a+b)*stirling1(i,a)*stirling1(n-i,b)/((b+1)*binomial(a+b+1,b+1)), b=1..n-i+1), a=1..i+1); end;
# Another program:
T:=proc(n, k) (-1)^(n-k)*(n/(n-1))*binomial(n-1, k-1)* integrate(expand(binomial(t-1, n))/(t-k), t=1..n); end;
[[1], seq( [seq(T(n, k), k=1..n)], n=2..14)];

a[n_, i_] /; i == 0 || i == n = 1/n! Sum[n^a*StirlingS1[n, a]/(a + 1), {a, 1, n + 1}]; a[n_, i_] = 1/n!*Binomial[n, i]*Sum[ n^(a + b)*StirlingS1[i, a]*StirlingS1[n - i, b]/((b + 1)*Binomial[a + b + 1, b + 1]), {b, 1, n}, {a, 1, i + 1}]; Table[a[n, i], {n, 0, 10}, {i, 0, n}] // Flatten // Numerator //  Take[#, 59]&
(* Jean-François Alcover, May 17 2011, after Maple prog. *)

A100621 Denominator of Cotesian number C(n,0).

Original entry on oeis.org

1, 2, 6, 8, 90, 288, 840, 17280, 28350, 89600, 598752, 17418240, 63063000, 402361344000, 5003856000, 295206912, 976924698750, 342372925440000, 15209113920000, 5377993912811520000, 96852084769440, 89903156428800000, 37556196837868800000, 73570956727261593600000

Offset: 0

N. J. A. Sloane, Dec 04 2004

			0, 1/2, 1/6, 1/8, 7/90, 19/288, 41/840, 751/17280, 989/28350, 2857/89600, 16067/598752, 434293/17418240, 1364651/63063000, 8181904909/402361344000, ... = A100620/A100621 = A002177/A002176 (the latter is not in lowest terms)

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
See A002176 for further references.

cn[n_, 0] := Sum[n^j*StirlingS1[n, j]/(j + 1), {j, 1, n + 1}]/n!; cn[n_, n_] := cn[n, 0]; cn[n_, k_] := 1/n!*Binomial[n, k]*Sum[n^(j + m)*StirlingS1[k, j]*StirlingS1[n - k, m]/((m + 1)*Binomial[j + m + 1, m + 1]), {m, 1, n}, {j, 1, k + 1}]; Table[cn[n, 0] // Denominator, {n, 0, 23}] (* Jean-François Alcover, Jan 16 2013 *)

A100620 Numerator of Cotesian number C(n,0).

Original entry on oeis.org

0, 1, 1, 1, 7, 19, 41, 751, 989, 2857, 16067, 434293, 1364651, 8181904909, 90241897, 5044289, 15043611773, 5026792806787, 203732352169, 69028763155644023, 1145302367137, 1022779523247467, 396760150748100749, 750218743980105669781, 35200969735190093

Offset: 0

N. J. A. Sloane, Dec 04 2004

			0, 1/2, 1/6, 1/8, 7/90, 19/288, 41/840, 751/17280, 989/28350, 2857/89600, 16067/598752, 434293/17418240, 1364651/63063000, 8181904909/402361344000, ... = A100620/A100621 = A002177/A002176 (the latter is not in lowest terms)

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.

See A002176 for further references. A diagonal of A100640/A100641.

(This defines the Cotesian numbers C(n,i)) with(combinat); C:=proc(n,i) if i=0 or i=n then RETURN( (1/n!)*add(n^a*stirling1(n,a)/(a+1),a=1..n+1) ); fi; (1/n!)*binomial(n,i)* add( add( n^(a+b)*stirling1(i,a)*stirling1(n-i,b)/((b+1)*binomial(a+b+1,b+1)), b=1..n-i+1), a=1..i+1); end;

cn[n_, 0] := Sum[n^j*StirlingS1[n, j]/(j + 1), {j, 1, n + 1}]/n!; cn[n_, n_] := cn[n, 0]; cn[n_, k_] := 1/n!*Binomial[n, k]*Sum[n^(j + m)*StirlingS1[k, j]*StirlingS1[n - k, m]/((m + 1)*Binomial[j + m + 1, m + 1]), {m, 1, n}, {j, 1, k + 1}]; Table[cn[n, 0] // Numerator, {n, 0, 24}] (* Jean-François Alcover, Jan 16 2013 *)

A321118 T(n,k) = A321119(n) - (-1)^kA321119(n-2k)/2 for 0 < k < n, with T(0,0) = 0 and T(n,0) = T(n,n) = A002530(n+1) for n > 0, triangle read by rows; unreduced numerator of the weights of Holladay-Sard's quadrature formula.

Original entry on oeis.org

0, 1, 1, 3, 10, 3, 4, 11, 11, 4, 11, 32, 26, 32, 11, 15, 43, 37, 37, 43, 15, 41, 118, 100, 106, 100, 118, 41, 56, 161, 137, 143, 143, 137, 161, 56, 153, 440, 374, 392, 386, 392, 374, 440, 153, 209, 601, 511, 535, 529, 529, 535, 511, 601, 209

Offset: 0

Franck Maminirina Ramaharo, Nov 01 2018

The n-th row common denominator is factorized out and is given by A321119(n).

Given a continuous function f over the interval [0,n], the best quadrature formula in the sense of Holladay-Sard is given by Integral_{x=0..n} f(x) dx = Sum_{k=0..n} T(n,k)*f(k)/A321119(n). The formula is exact if f belongs to the class of natural cubic splines.

			Triangle begins (denominator is factored out):
    0;                                                 1/4
    1,   1;                                            1/2
    3,  10,   3;                                       1/8
    4,  11,  11,   4;                                  1/10
   11,  32,  26,  32,  11;                             1/28
   15,  43,  37,  37,  43,  15;                        1/38
   41, 118, 100, 106, 100, 118,  41;                   1/104
   56, 161, 137, 143, 143, 137, 161,  56;              1/142
  153, 440, 374, 392, 386, 392, 374, 440, 153;         1/388
  209, 601, 511, 535, 529, 529, 535, 511, 601, 209;    1/530
  ...
If f is a continuous function over the interval [0,3], then the quadrature formula yields Integral_{x=0..3} f(x) d(x) = (1/10)*(4*f(0) + 11*f(1) + 11*f(2) + 4*f(3)).

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.2.

Franck Maminirina Ramaharo, Rows n = 0..150 of triangle, flattened
Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, Chapter II The Cubic Spline, Mathematics in Science and Engineering Volume 38 (1967), p. 9-74.
John C. Holladay, A smoothest curve approximation, Math. Comp. Vol. 11 (1957), 233-243.
Peter Köhler, On the weights of Sard's quadrature formulas, CALCOLO Vol. 25 (1988), 169-186.
Leroy F. Meyers and Arthur Sard, Best approximate integration formulas, J. Math. Phys. Vol. 29 (1950), 118-123.
Arthur Sard, Best approximate integration formulas; best approximation formulas, American Journal of Mathematics Vol. 71 (1949), 80-91.
Isaac J. Schoenberg, Spline interpolation and best quadrature formulae, Bull. Amer. Math. Soc. Vol. 70 (1964), 143-148.
Frans Schurer, On natural cubic splines, with an application to numerical integration formulae, EUT report. WSK, Dept. of Mathematics and Computing Science Vol. 70-WSK-04 (1970), 1-32.

Cf. A093735, A093736, A100641, A002176, A100640, A321119, A321120, A321121, A321122.

alpha = (Sqrt[2] + Sqrt[6])/2; T[0,0] = 0;
T[n_, k_] := If[n > 0 && k == 0 || k == n, (alpha^(n + 1) - (-alpha)^(-(n + 1)))/(2*Sqrt[6]*(alpha^n + (-alpha)^(-n))), 1 - (-1)^k*(alpha^(n - 2*k) + (-alpha)^(2*k - n))/(2*(alpha^n + (-alpha)^(-n)))];
a321119[n_] := 2^(-Floor[(n - 1)/2])*((1 - Sqrt[3])^n + (1 + Sqrt[3])^n);
Table[FullSimplify[a321119[n]*T[n, k]],{n, 0, 10}, {k, 0, n}] // Flatten

(b[0] : 0, b[1] : 1, b[2] : 1, b[3] : 3, b[n] := 4*b[n-2] - b[n-4])$ /* A002530 */
d(n) := 2^(-floor((n - 1)/2))*((1 - sqrt(3))^n + (1 + sqrt(3))^n) $ /* A321119 */
T(n, k) := if n = 0 and k = 0 then 0 else if n > 0 and k = 0 or k = n then b[n + 1] else d(n) - (-1)^k*d(n - 2*k)/2$
create_list(ratsimp(T(n, k)), n, 0, 10, k, 0, n);

T(n,k)/A321119(n) = (alpha^(n + 1) - (-alpha)^(-(n + 1)))/(2*sqrt(6)*(alpha^n + (-alpha)^(-n))) if k = 0 or k = n, and 1 - (-1)^k*(alpha^(n - 2*k) + (-alpha)^(2*k - n))/(2*(alpha^n + (-alpha)^(-n))) if 0 < k < n, where alpha = (sqrt(2) + sqrt(6))/2.
T(n,k) = T(n,n-k).
T(n,k) = 4*T(n-2,k) - T(n-4,k), n >= k + 4.
T(2*n+2,k)*A001834(n+1) = A001834(n)*T(2*n,k) + 2*A003500(n)*T(2*n+1,k) for k < 2*n.
T(2*n+3,k)*A003500(n+1) = A003500(n)*T(2*n+1,k) + 2*A001834(n+1)*T(2*n+2,k) for k < 2*n + 1.
Sum_{k=0..n} T(n,k)/A321119(n) = n.

A321119 a(n) = ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2); n-th row common denominator of A321118.

Original entry on oeis.org

4, 2, 8, 10, 28, 38, 104, 142, 388, 530, 1448, 1978, 5404, 7382, 20168, 27550, 75268, 102818, 280904, 383722, 1048348, 1432070, 3912488, 5344558, 14601604, 19946162, 54493928, 74440090, 203374108, 277814198, 759002504, 1036816702, 2832635908, 3869452610

Offset: 0

Franck Maminirina Ramaharo, Nov 01 2018

			a(0) = ((1 - sqrt(3))^0 + (1 + sqrt(3))^0)/2^floor((0 - 1)/2) = 2*(1 + 1) = 4.

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.2.

Encyclopedia of Mathematics, Quadrature formula
John C. Holladay, A smoothest curve approximation, Math. Comp. Vol. 11 (1957), 233-243.
Peter Köhler, On the weights of Sard's quadrature formulas, CALCOLO Vol. 25 (1988), 169-186.
Leroy F. Meyers and Arthur Sard, Best approximate integration formulas, J. Math. Phys. Vol. 29 (1950), 118-123.
Arthur Sard, Best approximate integration formulas; best approximation formulas, American Journal of Mathematics Vol. 71 (1949), 80-91.
Isaac J. Schoenberg, Spline interpolation and best quadrature formulae, Bull. Amer. Math. Soc. Vol. 70 (1964), 143-148.
Frans Schurer, On natural cubic splines, with an application to numerical integration formulae, EUT report. WSK, Dept. of Mathematics and Computing Science Vol. 70-WSK-04 (1970), 1-32.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Cf. A002176 (common denominators of Cotesian numbers).
Cf. A321118, A321120.
Cf. A001834, A002530, A002531, A003500, A005246, A048788, A083336, A125905.

LinearRecurrence[{0, 4, 0, -1}, {4, 2, 8, 10}, 50]

a(n) := ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2)$
makelist(ratsimp(a(n)), n, 0, 50);

a(n) = (((sqrt(2) - sqrt(6))/2)^n + ((sqrt(6) + sqrt(2))/2)^n)*((2 - sqrt(2))*(-1)^n + 2 + sqrt(2))/2.
a(-n) = (-1)^n*a(n).
a(n) = 2*A000034(n+1)*A002531(n).
a(2*n) = 2*A001834(n).
a(2*n+1) = 2*A003500(n).
a(n) = 4*a(n-2) - a(n-4) with a(0) = 4, a(1) = 2, a(2) = 8, a(3) = 10.
a(2*n+3) = a(2*n+1) + a(2*n+2).
a(2*n+2) = a(2*n) + 2*a(2*n+1).
G.f.: 2*(1 - x)*(2 + 3*x - x^2)/(1 - 4*x^2 + x^4).
E.g.f.: (1 + exp(-sqrt(6)*x))*((2 - sqrt(2))*exp(sqrt(2 - sqrt(3))*x) + (2 + sqrt(2))*exp(sqrt(2 + sqrt(3))*x))/2.
Lim_{n->infinity} a(2*n+1)/a(2*n) = (1 + sqrt(3))/2.

cp's OEIS Frontend

A002177 Numerators of Cotesian numbers (not in lowest terms): A002176(n)*C(n,0).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

Extensions

A002179 Numerators of Cotesian numbers (not in lowest terms): A002176*C(n,2).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A002178 Numerators of Cotesian numbers (not in lowest terms): A002176*C(n,1).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A100642 Triangle read by rows: numerators of Cotesian numbers C(n,k) (0 <= k <= n) if the denominators are set to the lcm's of the rows (A002176).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

A100641 Triangle read by rows: denominators of Cotesian numbers C(n,k) (0 <= k <= k).

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Maple

Mathematica

A100640 Triangle read by rows: numerators of Cotesian numbers C(n,k) (0 <= k <= n).

Views

Author

Keywords

References

Crossrefs

Programs

Maple

Mathematica

A100621 Denominator of Cotesian number C(n,0).

Views

Author

Keywords

Examples

References

Programs

Mathematica

A100620 Numerator of Cotesian number C(n,0).

Views

Author

A321118 T(n,k) = A321119(n) - (-1)^kA321119(n-2k)/2 for 0 < k < n, with T(0,0) = 0 and T(n,0) = T(n,n) = A002530(n+1) for n > 0, triangle read by rows; unreduced numerator of the weights of Holladay-Sard's quadrature formula.