A002176
a(n) = LCM of denominators of Cotesian numbers {C(n,k), 0 <= k <= n}.
Original entry on oeis.org
2, 6, 8, 90, 288, 840, 17280, 28350, 89600, 598752, 87091200, 63063000, 402361344000, 5003856000, 2066448384, 976924698750, 3766102179840000, 15209113920000, 5377993912811520000, 1646485441080480, 89903156428800000
Offset: 1
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 886.
- Louis Brand, Differential and Difference Equations, 1966, p. 612.
- W. W. Johnson, On Cotesian numbers: their history, computation and values to n=20, Quart. J. Pure Appl. Math., 46 (1914), 52-65.
- Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
- 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
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 886.
- 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]
-
Define C(n,k) as in A100640, then: A002176:=proc(n) local t1,k; t1:=1; for k from 0 to n do t1:=lcm(t1,denom(C(n,k))); od: t1; 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}]; a[n_] := LCM @@ Table[ Denominator[cn[n, k]], {k, 0, n}]; Table[a[n], {n, 1, 21}] (* 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
-
A002176(n)= denominator(cn(n))
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
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.
-
(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
- Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
- L. M. Milne-Thompson, Calculus of Finite Differences, MacMillan, 1951, p. 170.
-
(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. *)
A002177
Numerators of Cotesian numbers (not in lowest terms): A002176(n)*C(n,0).
Original entry on oeis.org
1, 1, 1, 7, 19, 41, 751, 989, 2857, 16067, 2171465, 1364651, 8181904909, 90241897, 35310023, 15043611773, 55294720874657, 203732352169, 69028763155644023, 19470140241329, 1022779523247467, 396760150748100749
Offset: 1
- 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).
-
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
-
ncn(n) = denominator(cn(n)) * cn(n);
-
nk(n,k) = if(k<0 || k>n, 0, ncn(n)[ k+1 ]);
-
A002177(n) = nk(n,0);
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
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 *)
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
- 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).
-
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)
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
- 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).
-
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)
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
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.
-
# (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 *)
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