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))
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);
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)
A100646
Denominator of Cotesian number C(n,2).
Original entry on oeis.org
6, 8, 15, 144, 280, 640, 14175, 2240, 199584, 87091200, 875875, 22353408000, 5003856000, 229605376, 10854718875, 941525544960000, 1013940928000, 3064383995904000, 82324272054024, 2996771880960000, 255484332230400000, 809280523999877529600000, 5699209469078125
Offset: 2
1/6, 3/8, 2/15, 25/144, 9/280, 49/640, -464/14175, 27/2240, -16175/199584, -3237113/87091200, -105387/875875, -1737125143/22353408000, -770720657/5003856000, -25881785/229605376, ... = A100645/A100646 = A002179/A002176 (the latter not being in lowest terms)
- Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
-
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_] := Denominator[cn[n, 2]]; Table[a[n], {n, 2, 24}] (* Jean-François Alcover, Oct 08 2013 *)
A100645
Numerator of Cotesian number C(n,2).
Original entry on oeis.org
1, 3, 2, 25, 9, 49, -464, 27, -16175, -3237113, -105387, -1737125143, -770720657, -25881785, -1997012608, -135505859252213, -214182958293, -528114253960241, -19467909708875, -595278405326437, -66462260889140083, -180690496141440384775397, -1610254561193224
Offset: 2
1/6, 3/8, 2/15, 25/144, 9/280, 49/640, -464/14175, 27/2240, -16175/199584, -3237113/87091200, -105387/875875, -1737125143/22353408000, -770720657/5003856000, -25881785/229605376, ... = A100645/A100646 = A002179/A002176 (the latter not being in lowest terms)
- Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
-
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_] := Numerator[cn[n, 2]]; Table[a[n], {n, 2, 24}] (* Jean-François Alcover, Oct 08 2013 *)
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