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Search: a278400 -id:a278400
Displaying 1-10 of 11 results found.
page 1 2
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A278420
a(n) = (A278399(n)^2 + A278400(n)^2)/2.
+20
7
1, 1, 1, 2, 2, 5, 4, 9, 10, 20, 18, 32, 45, 58, 82, 101, 148, 178, 274, 306, 452, 512, 785, 872, 1258, 1450, 2061, 2304, 3274, 3796, 5108, 6056, 7954, 9376, 12200, 14733, 18500, 22608, 28004, 34354, 41905, 51752, 62122, 77090, 91764, 114640, 134560, 167690
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
FORMULA
a(n) = A292042(n)^2 + A292043(n)^2. - Vaclav Kotesovec, Sep 08 2017
MATHEMATICA
Abs[(QPochhammer[I, x] + O[x]^60)[[3]]]^2 / 2
CROSSREFS
Cf. A278399, A278400, A292042, A292043.
KEYWORD
nonn
AUTHOR
Vladimir Reshetnikov, Nov 21 2016
STATUS
approved
A278399
G.f.: Re((i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).
+10
12
1, -1, -1, -2, -2, -3, -2, -3, -2, -2, 0, 0, 3, 4, 8, 9, 14, 16, 22, 24, 30, 32, 39, 40, 46, 46, 51, 48, 52, 46, 46, 36, 32, 16, 8, -15, -30, -60, -82, -122, -151, -200, -238, -296, -342, -412, -464, -542, -602, -686, -750, -841, -904, -996, -1058, -1146, -1198
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
The q-Pochhammer symbol (a; q)_inf = Product_{k>=0} (1 - a*q^k).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
FORMULA
(i; x)_inf is the g.f. for a(n) + i*A278400(n).
G.f.: Sum_{n >= 0} (-1)^n^x^(n*(2*n-1))/Product_{k = 1..2*n} 1 - x^k. - Peter Bala, Feb 04 2021
MAPLE
G := add((-1)^n*x^(n*(2*n-1))/mul(1 - x^k, k = 1..2*n), n = 0..7):
S := series(G, x, 101):
seq(coeff(S, x, j), j = 0..100); # Peter Bala, Feb 04 2021
MATHEMATICA
Re[(QPochhammer[I, x] + O[x]^60)[[3]]]
CROSSREFS
Cf. A278400, A278401, A278402, A278420, A292042, A292043.
KEYWORD
sign,easy
AUTHOR
Vladimir Reshetnikov, Nov 20 2016
STATUS
approved
A292043
G.f.: Im((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).
+10
10
0, -1, -1, -1, -1, -1, 0, 0, 1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 15, 15, 16, 16, 16, 14, 13, 9, 6, 0, -5, -14, -22, -34, -45, -60, -74, -93, -110, -132, -152, -177, -199, -226, -249, -277, -300, -328, -348, -373, -389, -408, -417, -428, -425, -424, -407, -389, -352
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,10
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
FORMULA
(i*x; x)_inf is the g.f. for A292042(n) + i*a(n).
A292042(n)^2 + a(n)^2 = A278420(n). - Vaclav Kotesovec, Sep 08 2017
From Peter Bala, Feb 05 2021: (Start)
G.f.: Sum_{n >= 0} (-1)^(n+1)*x^((n+1)*(2*n+1))/Product_{k = 1..2*n+1} (1 - x^k).
The 2 X 2 matrix Product_{k >= 1} [1, -x^k; x^k, 1] = [A(x), B(x); -B(x), A(x)], where A(x) is the g.f. of A292042 and B(x) is the g.f. for this sequence.
A(x)^2 + B(x)^2 = Product_{k >= 1} 1 + x^(2*k) = A000009(x^2).
A(x) + B(x) is the g.f. of A278399; B(x) - A(x) is the g.f. of A278400. (End)
EXAMPLE
Product_{k>=1} (1 - i*x^k) = 1 + (0-1i)*x + (0-1i)*x^2 + (-1-1i)*x^3 + (-1-1i)*x^4 + (-2-1i)*x^5 + (-2+0i)*x^6 + (-3+0i)*x^7 + ...
MAPLE
N:= 100:
S := convert(series( add( (-1)^(n+1)*x^((n+1)*(2*n+1))/(mul(1 - x^k, k = 1..2*n+1)), n = 0..floor(sqrt(N/2)) ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Feb 05 2021
MATHEMATICA
Im[CoefficientList[Series[QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 08 2017 *)
CROSSREFS
Cf. A278399, A278400, A278420, A292042, A292052.
KEYWORD
sign
AUTHOR
Seiichi Manyama, Sep 08 2017
STATUS
approved
A292042
G.f.: Re((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).
+10
9
1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -3, -4, -3, -3, -1, -1, 2, 3, 7, 9, 14, 16, 23, 26, 33, 37, 45, 48, 57, 60, 68, 70, 77, 76, 82, 78, 80, 72, 70, 55, 48, 26, 11, -19, -42, -84, -116, -169, -213, -278, -333, -413, -479, -572, -651, -757, -846, -965, -1062
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
FORMULA
( i*x; x)_inf is the g.f. for a(n) + i*A292043(n).
(-i*x; x)_inf is the g.f. for a(n) + i*A292052(n).
a(n)^2 + A292043(n)^2 = A278420(n). - Vaclav Kotesovec, Sep 08 2017
From Peter Bala, Jan 15 2021: (Start)
G.f.: A(x) = Sum_{n >= 0} (-1)^n*x^(n*(2*n+1))/Product_{k = 1..2*n} (1 - x^k). Cf. A035294.
Conjectural g.f.: A(x) = (1/2)*Sum_{n >= 0} (-x)^(n*(n-1)/2)/Product_{k = 1..n} (1 - x^k). (End)
EXAMPLE
Product_{k>=1} (1 - i*x^k) = 1 + (0-1i)*x + (0-1i)*x^2 + (-1-1i)*x^3 + (-1-1i)*x^4 + (-2-1i)*x^5 + (-2+0i)*x^6 + (-3+0i)*x^7 + ...
MAPLE
N:= 100:
S := convert(series( add( (-1)^n*x^(n*(2*n+1))/(mul(1 - x^k, k = 1..2*n)), n = 0..floor(sqrt(N/2)) ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021
MATHEMATICA
Re[CoefficientList[Series[QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 08 2017 *)
CROSSREFS
Cf. A035294, A278399, A278400, A278420, A292043, A292052.
KEYWORD
sign
AUTHOR
Seiichi Manyama, Sep 08 2017
STATUS
approved
A278401
G.f.: Re(2/(i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).
+10
5
1, -1, -2, -1, -1, -1, -1, 1, 2, 2, 2, 4, 5, 5, 5, 6, 7, 5, 3, 4, 3, 0, -2, -3, -5, -10, -14, -16, -18, -23, -28, -28, -29, -35, -38, -37, -37, -39, -39, -35, -30, -29, -26, -15, -5, 0, 10, 26, 41, 51, 64, 85, 105, 119, 135, 160, 183, 196, 212, 236, 255, 265
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The q-Pochhammer symbol (a; q)_inf = Product_{k>=0} (1 - a*q^k).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
FORMULA
2/(i; x)_inf is the g.f. for a(n) + i*A278402(n).
G.f.: Sum_{n >= 0} (-1)^n*x^(2*n)*(1 - x - x^(2*n+1))/Product_{k = 1..2*n+1} (1 - x^k). - Peter Bala, Feb 08 2021
MAPLE
with(gfun): series( add( (-1)^n*x^(2*n)*(1 - x - x^(2*n+1))/mul(1 - x^k, k = 1..2*n+1), n = 0..50), x, 101): seriestolist(%); # Peter Bala, Feb 08 2021
MATHEMATICA
Re[(2/QPochhammer[I, x] + O[x]^70)[[3]]]
CROSSREFS
Cf. A278399, A278400, A278402.
KEYWORD
sign,easy
AUTHOR
Vladimir Reshetnikov, Nov 20 2016
STATUS
approved
A278402
G.f.: Im(2/(i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).
+10
5
1, 1, 0, -1, -1, -1, -3, -3, -2, -2, -2, -2, -1, 1, 1, 2, 5, 7, 7, 8, 11, 12, 12, 13, 15, 16, 14, 12, 12, 11, 6, 2, 1, -3, -10, -17, -21, -27, -37, -45, -50, -57, -68, -77, -81, -86, -96, -102, -101, -103, -108, -109, -103, -97, -95, -88, -71, -54, -42, -24, 5
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
The q-Pochhammer symbol (a; q)_inf = Product_{k>=0} (1 - a*q^k).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
FORMULA
2/(i; x)_inf is the g.f. for A278401(n) + i*a(n).
G.f.: Sum_{n >= 0} (-1)^n*x^(2*n)*(1 + x - x^(2*n+1))/Product_{k = 1..2*n+1} (1 - x^k). - Peter Bala, Feb 09 2021
MAPLE
with(gfun): series( add( (-1)^n*x^(2*n)*(1 + x - x^(2*n+1))/mul(1 - x^k, k = 1..2*n+1), n = 0..50), x, 101): seriestolist(%); # Peter Bala, Feb 09 2021
MATHEMATICA
Im[(2/QPochhammer[I, x] + O[x]^70)[[3]]]
CROSSREFS
Cf. A278399, A278400, A278401.
KEYWORD
sign
AUTHOR
Vladimir Reshetnikov, Nov 20 2016
STATUS
approved
A292052
G.f.: Im((-i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).
+10
4
0, 1, 1, 1, 1, 1, 0, 0, -1, -2, -3, -4, -6, -7, -9, -10, -12, -13, -15, -15, -16, -16, -16, -14, -13, -9, -6, 0, 5, 14, 22, 34, 45, 60, 74, 93, 110, 132, 152, 177, 199, 226, 249, 277, 300, 328, 348, 373, 389, 408, 417, 428, 425, 424, 407, 389, 352, 314, 252
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,10
LINKS
Robert Israel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
FORMULA
(-i*x; x)_inf is the g.f. for A292042(n) + i*a(n).
a(n) = -A292043(n).
EXAMPLE
Product_{k>=1} (1 + i*x^k) = 1 + (0+1i)*x + (0+1i)*x^2 + (-1+1i)*x^3 + (-1+1i)*x^4 + (-2+1i)*x^5 + (-2+0i)*x^6 + (-3+0i)*x^7 + ...
MAPLE
N:= 100: # to get a(0)..a(N)
P:= mul(1+I*x^k, k=1..N):
S:= series(P, x, N+1):
seq(evalc(Im(coeff(S, x, j))), j=0..N); # Robert Israel, Sep 08 2017
MATHEMATICA
Im[CoefficientList[Series[QPochhammer[-I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 09 2017 *)
CROSSREFS
Cf. A278399, A278400, A278420, A292042, A292043.
KEYWORD
sign,look
AUTHOR
Seiichi Manyama, Sep 08 2017
STATUS
approved
A292140
G.f.: Im((2*i; x)_oo), where (a; q)_oo is the q-Pochhammer symbol, i = sqrt(-1).
+10
4
-2, -2, -2, 6, 6, 14, 22, 30, 38, 54, 30, 46, 30, 14, -34, -74, -154, -226, -362, -498, -698, -762, -1058, -1218, -1474, -1634, -1890, -1914, -2074, -2002, -1962, -1570, -1210, -266, 606, 2190, 3454, 6030, 8382, 11926, 15334, 20190, 24758, 30990, 36678, 44134
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
FORMULA
(2*i; x)_oo is the g.f. for A292135(n) + i*a(n).
CROSSREFS
Column k=2 of A292160.
Cf. A278400, A292135, A292141.
KEYWORD
sign
AUTHOR
Seiichi Manyama, Sep 09 2017
STATUS
approved
A292049
Triangle read by rows: T(n,k) = (-1)^(k-1) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).
+10
3
1, 0, 1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, -2, 0, 1, -2, -1, 0, 1, -3, -1, 0, 1, -3, -2, 0, 1, -4, -3, 0, 1, -4, -4, 1, 0, 1, -5, -5, 1, 0, 1, -5, -7, 2, 0, 1, -6, -8, 3, 0, 1, -6, -10, 5, 0, 1, -7, -12, 6, 1, 0, 1, -7, -14, 9, 1, 0, 1, -8, -16, 11, 2, 0, 1, -8, -19
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,14
LINKS
Seiichi Manyama, Rows n = 0..481, flattened
EXAMPLE
First few rows are:
1;
0, 1;
0, 1;
0, 1, -1;
0, 1, -1;
0, 1, -2;
0, 1, -2, -1;
0, 1, -3, -1;
0, 1, -3, -2;
0, 1, -4, -3;
0, 1, -4, -4, 1.
CROSSREFS
Row sums give (-1)*A278400.
Columns 0-1 give A000007, A000012.
Cf. A292047.
KEYWORD
sign,tabf,look
AUTHOR
Seiichi Manyama, Sep 08 2017
STATUS
approved
A292160
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Im((k*i; x)_inf), and (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).
+10
3
0, -1, 0, -2, -1, 0, -3, -2, -1, 0, -4, -3, -2, 0, 0, -5, -4, -3, 6, 0, 0, -6, -5, -4, 24, 6, 1, 0, -7, -6, -5, 60, 24, 14, 2, 0, -8, -7, -6, 120, 60, 51, 22, 3, 0, -9, -8, -7, 210, 120, 124, 78, 30, 4, 0, -10, -9, -8, 336, 210, 245, 188, 105, 38, 6, 0, -11, -10
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Table of n, a(n) for n=0..67.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
EXAMPLE
Square array begins:
0, -1, -2, -3, -4, ...
0, -1, -2, -3, -4, ...
0, -1, -2, -3, -4, ...
0, 0, 6, 24, 60, ...
0, 0, 6, 24, 60, ...
CROSSREFS
Columns k=0..2 give A000004, A278400, A292140.
Rows 0+2 give (-1)*A001477.
Main diagonal gives A292162.
Cf. A292159.
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Sep 10 2017
STATUS
approved
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