A003406 -id:A003406 - cp's OEIS frontend

A158690 Expansion of the basic hypergeometric series 1 + (1 - exp(-t)) + (1 - exp(-t))(1 - exp(-3t)) + (1 - exp(-t))(1 - exp(-3t))(1 - exp(-5t)) + ... as a series in t.

1, 1, 5, 55, 1073, 32671, 1431665, 85363615, 6646603073, 654896692351, 79656194515025, 11722538113191775, 2052949879753739873, 421931472111868912831, 100568330857984368195185

Offset: 0

Peter Bala, Mar 24 2009

We appear to get the same sequence by expanding 1 - (1 - exp(t)) + (1 - exp(t))*(1 - exp(2*t)) - (1 - exp(t))*(1 - exp(2*t))*(1 - exp(3*t)) + ... as a series in t. Compare with A079144. For other sequences with generating functions of a similar type see A000364, A000464, A002105 and A002439.
From Peter Bala, Mar 13 2017: (Start)
It appears that the g.f. has two other forms: either F(exp(-t)) where F(q) = Sum_{n >= 0} q^(n+1)*Product_{k = 1..n} 1 - q^(2*k) = q + q^2 + q^3 - q^7 - q^8 - q^10 - q^11 - ... is a g.f. for A003475 or 1/2*G(exp(t)) where G(q) = 1 + Sum_{n >= 0} (-1)^n*q^(n+1)*Product_{k = 1..n} 1 - q^k = 1 + q - q^2 + 2*q^3 - 2*q^4 + q^5 + q^7 - 2*q^8 + ... is a g.f. for A003406. See Zagier, Example 1. (End)
From Peter Bala, Dec 18 2021: (Start)
Conjectures:
1) Taking the sequence modulo an integer k gives an eventually periodic sequence with period dividing phi(k). For example, the sequence taken modulo 16 begins [1, 1, 5, 7, 1, 15, 1, 15, 1, 15, 1, 15, ...] with an apparent pre-period of length 4 and a period of length 2.
2) Let i >= 0 and define a_i(n) = a(n+i). Then for each i the Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k.
If true, then for each i the expansion of exp( Sum_{n >= 1} a_i(n)*x^n/n ) has integer coefficients. For example, the expansion of exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 21*x^3 + 291*x^4 + 6861*x^5 + 246171*x^6 + 12458901*x^7 + 843915891*x^8 + 73640674461*x^9 + 8041227405771*x^10 + ... appears to have integer coefficients. (End)

			G.f. A(x) = 1 + x + 5*x^2 + 55*x^3 + 1073*x^4 + 32671*x^5 + 1431665*x^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..170
Hsien-Kuei Hwang, and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.
D. Zagier, Quantum modular forms, Quanta of Maths: Conference in honor of Alain Connes, Clay Mathematics Proceedings 11, AMS and Clay Mathematics Institute 2010, 659-675

Cf. A000364, A000464, A002105, A002439, A079144, A158691, A209832, A215066, A003406, A003475.

max = 14; se = Series[1 + Sum[ Product[1 - E^(-(2*k - 1)*t), {k, 1, n}], {n, 1, max}], {t, 0, max}]; CoefficientList[se, t]*Range[0, max]! (* Jean-François Alcover, Mar 06 2013 *)

{a(n)=n!*polcoeff(sum(m=0, n, prod(k=1, m, 1-exp(-(2*k-1)*x+x*O(x^n)))), n)} \\ Paul D. Hanna, Aug 01 2012

{a(n)=n!*polcoeff(sum(m=0, n, prod(k=1, m, exp(k*x+x*O(x^n))-1)), n)} \\ Paul D. Hanna, Aug 01 2012

Basic hypergeometric generating function: 1 + Sum_{n >= 0} Product_{k = 1..n} (1 - exp(2*k-1)*t) = 1 + t + 5*t^2/2! + 55*t^3/3! + ....
a(n) ~ 6*sqrt(2) * 12^n * (n!)^2 / Pi^(2*n+2). - Vaclav Kotesovec, May 04 2014
Conjectural g.f.: G(exp(t)) as a formal power series in t, where G(q) := Sum_{n >= 0} q^(2*n+1) * Product_{k = 1..2*n} (1 - q^k). - Peter Bala, May 16 2017
E.g.f.: Sum_{n>=0} exp(n*(n+1)/2*x) / Product_{k=0..n} (1 + exp(k*x)). - Paul D. Hanna, Oct 14 2020

A117193 Number of partitions of n into distinct parts with an odd rank.

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 2, 4, 4, 4, 6, 8, 10, 10, 13, 16, 20, 22, 28, 32, 38, 43, 52, 62, 72, 82, 96, 110, 128, 148, 170, 196, 224, 256, 292, 334, 380, 432, 490, 557, 630, 714, 806, 908, 1022, 1152, 1294, 1456, 1632, 1830, 2049, 2290, 2560, 2860, 3188, 3554, 3958, 4404

Offset: 1

Reinhard Zumkeller, Mar 03 2006

nonn

Cf. A000009, A003406, A101707, A117192.

a[n_] := Count[IntegerPartitions[n], q_ /; OddQ[First[q] - Length[q]] && Length[q] == Length[Union[q]]];
Array[a, 60] (* Jean-François Alcover, Oct 06 2021 *)

a(n) = Sum(A117195(n,k)*(k mod 2): 0<=k

a(n) = A000009(n) - A117192(n).

a(n) = A117192(n) - A003406(n).

A117192 Number of partitions of n into distinct parts with an even rank.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 3, 2, 4, 6, 6, 7, 8, 12, 14, 16, 18, 24, 26, 32, 38, 46, 52, 60, 70, 83, 96, 112, 128, 148, 170, 194, 224, 256, 293, 334, 380, 432, 492, 556, 630, 712, 804, 908, 1026, 1152, 1296, 1454, 1632, 1828, 2048, 2292, 2560, 2858, 3190, 3554, 3959, 4404

Offset: 1

Reinhard Zumkeller, Mar 03 2006

nonn

Cf. A000009, A003406, A101709, A101708, A117193, A117194, A117195.

a[n_] := Count[IntegerPartitions[n], q_ /; EvenQ[First[q] - Length[q]] && Length[q] == Length[Union[q]]];
Array[a, 60] (* Jean-François Alcover, Oct 06 2021 *)

a(n) = Sum(A117195(n,k)*(1 - k mod 2): 0<=k

a(n) = A117194(n)+A010054(n) = A000009(n)-A117193(n) = A117193(n)+A003406(n).

A005895 Weighted count of partitions with distinct parts.

Original entry on oeis.org

1, 2, 5, 7, 12, 18, 26, 35, 50, 67, 88, 116, 149, 191, 245, 306, 381, 477, 585, 718, 880, 1067, 1288, 1555, 1863, 2226, 2656, 3151, 3726, 4406, 5180, 6077, 7124, 8316, 9691, 11278, 13080, 15146, 17517, 20204, 23264, 26759, 30705, 35182, 40274, 46000, 52473, 59795, 68018, 77279, 87711, 99395, 112508

Offset: 1

N. J. A. Sloane and Simon Plouffe

Also sum of largest parts of all partitions of n into distinct parts. - Vladeta Jovovic, Feb 15 2004

Andrews, George E.; Ramanujan's "lost" notebook. V. Euler's partition identity. Adv. in Math. 61 (1986), no. 2, 156-164.
S.-Y. Kang, Generalizations of Ramanujan's reciprocity theorem..., J. London Math. Soc., 75 (2007), 18-34. See Eq. (1.5) but beware errors.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A005896, A003406, A336902, A336903.

M:=201; add( mul( (1+q^j),j=1..M) - mul( (1+q^j),j=1..n), n=0..M);
# second Maple program:
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(
      n=0, 1, b(n,i-1)+`if`(i>n, 0, b(n-i, min(n-i,i-1)))))
    end:
a:= n-> add(j*b(n-j, min(n-j,j-1)), j=1..n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 03 2016

m = 46; f[q_] :=  Sum[ Product[ (1+q^j), {j, 1, m}] - Product[ (1+q^j), {j, 1, n}], {n, 0, m}]; CoefficientList[ f[q], q][[2 ;; m+1]] (* Jean-François Alcover, Apr 13 2012, after Maple *)

N=66;  x='x+O('x^N);
S=prod(k=1,N, 1+x^k); gf=sum(n=0,N, S-prod(k=1,n, 1+x^k));
/* alternative: Arndt's g.f.: */
/* gf=sum(k=0,N, (k+1)*x^(k+1) * prod(j=1,k, 1+x^j) ); */
Vec(gf)
/* Joerg Arndt, Sep 17 2012 */

G.f.: sum(n>=0, S(q) - prod(k=1..n, 1+q^k) ), where S(q)=prod(k>=1, 1+q^k) (g.f. for A000009).

G.f. sum(k>=0, (k+1)*x^(k+1) * prod(j=1..k, 1+x^j) ). [Joerg Arndt, Sep 17 2012]

More terms from James Sellers, Dec 24 1999

A003475 Expansion of Sum_{k>0} (-1)^(k+1) q^(k^2) / ((1-q)(1-q^3)(1-q^5)...(1-q^(2k-1))).

Original entry on oeis.org

1, 1, 1, 0, 0, 0, -1, -1, 0, -1, -1, 0, -1, 0, 1, -1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 1, -1, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, -1, -1, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 1, 0, 1, 1, 0, 1, -1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, 0, 0, 0, -1, -2, 0, 0, -1, 0, 1, 0, -1, -1, -1, 0, 0, 0

Offset: 1

N. J. A. Sloane

sign

|a(n)|<3 if n<1036, a(1036)=3. - Michael Somos, Sep 16 2006

			G.f. = x + x^2 + x^3 - x^7 - x^8 - x^10 - x^11 - x^13 + x^15 - x^16 + ...

F. J. Dyson, A walk through Ramanujan's garden, pp. 7-28 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
F. J. Dyson, Selected Papers, Am. Math. Soc., 1996, p. 204.

Robert Israel, Table of n, a(n) for n = 1..2500
G. E. Andrews, F. J. Dyson and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988) 391-407.
G. E. Andrews, F. G. Garvan, and J. Liang, Self-conjugate vector partitions and the parity of the spt-function, Acta Arithmetica Vol. 158, Issue 3: 199-218 (2013) doi.org/10.4064/aa158-3-1
Alexander E. Patkowski, A note on the rank parity function, Discrete Math. 310 (2010), 961-965.
D. Zagier, Quantum modular forms, Example 1 in Quanta of Maths: Conference in honor of Alain Connes, Clay Mathematics Proceedings 11, AMS and Clay Mathematics Institute 2010, 659-675

Cf. A003406, A053251, A158690.

P:=n->mul((1-q^(2*i+1)),i=0..n-1):
t5:=add((-1)^(n+1)*q^(n^2)/P(n),n=1..40):
t6:=series(t5,q,40); # Based on Patkowski, 2010, Eq. 3.1. - N. J. A. Sloane, Jun 29 2011

a[ n_] := SeriesCoefficient[ 1 - QHypergeometricPFQ[ {x^2}, {x}, x^2, x], {x, 0, n}]; (* Michael Somos, Feb 02 2015 *)

{a(n) = local(A, p, e, x, y); if( n<0, 0, n = 24*n-1; A = factor(n); prod(k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p<5, 0, if( p%24>1 && p%24<23, if( e%2, 0, if( p%24==7 || p%24==17, (-1)^(e/2), 1)), x=y=0; if( p%24==1, forstep(i=1, sqrtint(p), 2, if( issquare( (i^2 + p) / 2, &y), x=i; break)), for(i=1, sqrtint(p\2), if( issquare( 2*i^2 + p,&x), y=i; break))); (e+1) * (-1)^( (x + if((x-y)%6, y, -y)) / 6*e))))) / -2)}; /* Michael Somos, Aug 17 2006 */

{a(n) = local(A); if( n<1, 0, A = -1 + x * O(x^n); polcoeff( sum(k=1, sqrtint(n), A *= 1 / (1 - x^(1 - 2*k)) * (1 + x * O(x^(n - k^2)))), n))}; /* Michael Somos, Sep 16 2006 */

Define c(24*k + 1) = A003406(k), c(24*k - 1) = -2*A003475(k), c(n) = 0 otherwise. Then c(n) is multiplicative with c(2^e) = c(3^e) = 0^e, c(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 7, 17 (mod 24), c(p^e) = (1 + (-1)^e) / 2 if p == 5, 11, 13, 19 (mod 24), c(p^e) = (e+1)*(-1)^(y*e) where p == 1, 23 (mod 24) and p = x^2 - 72*y^2 . - Michael Somos, Aug 17 2006

G.f.: x + x^2 * (1 - x^2) + x^3 * (1 - x^2) * (1 - x^4) + x^4 * (1 - x^2) * (1 - x^4) * (1 - x^6) + ... . - Michael Somos, Aug 18 2006

A291954 Triangle read by rows: T(n,k) = T(n-k,k-1) - T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, -1, 0, 1, 1, 0, -1, -1, 0, 1, 0, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, -1, 0, 0, 1, 0, -1, 0, -1, 0, 2, 1, 0, 1, 1, -1, -1, 0, -1, -1, 1, 0, 0, 1, 0, -2, -1, 0, -1, 0, 2, 1, 0, 1, 1, -2, 0, 1, 0, -1, -1, 2, 1, -1, 0, 1, 0, -2, -1, 0, 0, -1, 0, 3, 1, -1, 0, 1

Offset: 0

Seiichi Manyama, Sep 06 2017

			First few rows are:
  1;
  0,  1;
  0, -1;
  0,  1,  1;
  0, -1, -1;
  0,  1,  0;
  0, -1,  0,  1;
  0,  1,  1, -1;
  0, -1, -1,  0;
  0,  1,  0, -1;
  0, -1,  0,  2, 1.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A003406.
Columns 0-1 give A000007, A062157.
Cf. A008289, A291895, A291929.

G.f. of column k: x^(k*(k+1)/2) / Product_{j=1..k} (1+x^j).

A203568 a(n) = A026837(n) - A026838(n).

Original entry on oeis.org

0, 1, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0

Offset: 0

Michael Somos, Jan 03 2012

sign

			G.f. = x - x^2 + x^5 - x^7 + x^12 - x^15 + x^22 - x^26 + x^35 - x^40 + x^51 - ...
G.f. = q^25 - q^49 + q^121 - q^169 + q^289 - q^361 + q^529 - q^625 + ..
From _Peter Bala_, Feb 13 2020: (Start)
G.f.s for the tails of A(x):
Sum_{n >= 1} (-1)^(n+1) * x^(2*n+3)*Product_{k = 2..n} 1 + x^k = x^5 - x^7 + x^12 - x^15 + x^22 - ....
Sum_{n >= 2} (-1)^n * x^(3*n+6)*Product_{k = 3..n} 1 + x^k = x^12 - x^15 + x^22 - x^26 + x^35 - ....
Sum_{n >= 3} (-1)^(n+1) * x^(4*n+10)*Product_{k = 4..n} 1 + x^k =
x^22 - x^26 + x^35 - x^40 + x^51 - .... (End)

Robert Israel, Table of n, a(n) for n = 0..10000
George E. Andrews, Euler's Pentagonal Number Theorem, Mathematics Magazine, Vol. 56, No. 5 (Nov., 1983), pp. 279-284.

Cf. A003406, A010815, A026837, A026838, A143062.

N:= 1000: # to get a(0) to a(N)
V:= Array(0..N):
for k from 1 to floor((sqrt(1+24*N)-1)/6) do V[(3*k^2-k)/2]:= 1 od:
for k from 1 to floor((sqrt(1+24*N)+1)/6) do V[(3*k^2+k)/2]:= -1 od:
convert(V,list); # Robert Israel, Nov 24 2015

a[ n_] := Which[ n < 1, 0, SquaresR[ 1, 24 n + 1] == 2, -(-1)^Quotient[ Sqrt[24 n + 1], 3], True, 0]; (* Michael Somos, Jul 12 2015 *)

{a(n) = if( n<1, 0, if( issquare( 24*n + 1, &n), - kronecker( -12, n)))};

G.f.: Sum_{k in Z} sign(k) * x^(k * (3*k - 1) / 2).
G.f.: Sum_{k>0} x^(k * (3*k - 1) / 2) * (1 - x^k). - Michael Somos, Jul 12 2015
G.f.: x - x^2 * (1 + x) + x^3 * (1 + x) * (1 + x^2) - x^4 * (1 + x) * (1 + x^2) * (1 + x^3) + .... - Michael Somos, Jul 12 2015
G.f.: x / (1 + x) - x^3 / ((1 + x) * (1 + x^2)) + x^6 / ((1 + x) * (1 + x^2) * (1 + x^3)) - .... - Michael Somos, Jul 12 2015
G.f.: x / (1 + x^2) - x^2 / ((1 + x^2) * (1 + x^4)) + x^3 / ((1 + x^2 ) * (1 + x^4) * (1 + x^6)) - .... - Michael Somos, Jul 12 2015
a(n) = - A143062(n) unless n=0. - Michael Somos, Jul 12 2015
For k >= 1, a((3*k^2 - k)/2) = 1, a((3*k^2 + k)/2) = -1.  a(n) = 0 otherwise. - Robert Israel, Nov 24 2015
From Peter Bala, Feb 11 2021: (Start)
G.f.: A(x) = Sum_{n >= 1} x^(n*(2*n-1))/Product_{k = 1..2*n} 1 + x^k = x - x^2 + x^5 - x^7 + x^12 - x^15 + - ..., follows by adding terms in pairs in the above g.f. Sum_{n >= 1} (-1)^(n+1)*x^(n*(n+1)/2)/Product_{k = 1..n} 1 + x^k of Somos, dated Jul 12 2015.
G.f.: A(x) = 1/2 + (1/2)*Sum_{n >= 1} (-1)^n*x^(n*(n-1)/2)/Product_{k = 1..n} 1 + x^k.
A(x) = Sum_{n >= 0} (-1)^n * x^(n+1)*Product_{k = 1..n} 1 + x^k. (Set x = -1 in Andrews, equation 8. For similar results see the Examples below.)
Conjectural g.f: A(x) = Sum_{n >= 1} (-1)^(n+1) * x^(2*n-1)/Product_{k = 1..n} 1 + x^(2*k-1)  = x - x^2 + x^5 - x^7 + x^12 - x^15 + - ....
More generally, for positive integer N, we appear to have the identity
A(x) = Product_{j = 1..N-1} 1/(1 + x^(2*j)) * ( P(N,x) + Sum_{n >= 1} (-1)^(n+N) * x^(2*N*n-N)/Product_{k = 1..n} 1 + x^(2*k-1) ), where P(N,x) is a polynomial in x of degree N^2 - N - 1 for N > 1, with the first few values given empirically by P(1,x) = 0, P(2,x) = x, P(3,x) = x - x^2 + x^5 and P(4,x) = x - x^2 + x^3 + x^5 + x^7 - x^8 + x^11. Cf. A186424. (End)

A005896 Weighted count of partitions with odd parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 7, 9, 14, 19, 26, 34, 45, 59, 76, 96, 121, 153, 189, 234, 288, 353, 428, 519, 625, 752, 900, 1073, 1274, 1512, 1784, 2101, 2470, 2894, 3382, 3946, 4590, 5330, 6179, 7144, 8246, 9505, 10931, 12552, 14396, 16476, 18831, 21495

Offset: 0

N. J. A. Sloane, Simon Plouffe

			G.f. = x^3 + x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 9*x^8 + 14*x^9 + 19*x^10 + ... - _Michael Somos_, Oct 21 2018

Andrews, George E. Ramanujan's "lost" notebook. V. Euler's partition identity. Adv. in Math. 61 (1986), no. 2, 156-164.
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 = 0..300

Cf. A000009, A005895, A003406.

max = 48; f[n_, x_] := Product[ 1/(1-x^(2k+1)), {k, 0, n}]; g[x_] = Sum[ f[max/2, x] - f[n, x], {n, 0, max/2}]; CoefficientList[ Series[ g[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 17 2011, after g.f. *)
a[ n_] := With[{A = 1 / QPochhammer[ q, q^2]}, SeriesCoefficient[ Sum[A - 1 / QPochhammer[ q, q^2, k], {k, 1, n/2}], {q, 0, n}]]; (* Michael Somos, Oct 21 2018 *)

/* set maximum */ MM = 50; /* G.f. for partitions with odd parts: */ (Q(n, q) = prod(k=0, n, 1/(1 - q^(2*k+1)), 1 + q*O(q^MM))); /* G.f. for A000009: */ Sq = Q(MM/2, q); /* G.f. for A005896: */ Sq0 = sum(n=0, MM/2, Sq-Q(n, q)); for(n=0, 48, print1(polcoeff(Sq0, n)","));

G.f.: Sum_{n=0..infinity} {S(q)-1/((1-q)(1-q^3)...(1-q^(2n+1)))}, where S(q) = g.f. for A000009.

More terms from Michael Somos.

A121372 Triangle, read by rows of length A003056(n) for n >= 1, defined by the recurrence: T(n,k) = T(n-k,k-1) - T(n-k,k) for n > k > 1, with T(n,1) =(-1)^(n-1) for n >= 1.

Original entry on oeis.org

1, -1, 1, 1, -1, -1, 1, 0, -1, 0, 1, 1, 1, -1, -1, -1, 0, 1, 0, -1, -1, 0, 2, 1, 1, 1, -1, -1, -1, -1, 1, 0, 1, 0, -2, -1, -1, 0, 2, 1, 1, 1, -2, 0, 1, -1, -1, 2, 1, -1, 1, 0, -2, -1, 0, -1, 0, 3, 1, -1, 1, 1, -3, -2, 1, -1, -1, 2, 1, -1, 1, 0, -3, -1, 2, 1, -1, 0, 4, 2, -1, -1, 1, 1, -3, -1, 2, 0, -1, -1, 3, 1, -3, -1, 1, 0, -4, -2, 2, 1, -1, 0, 4, 2, -3

Offset: 1

Paul D. Hanna, Jul 24 2006

Row sums equal A003406 (offset 1), the expansion of Ramanujan's function: R(x) = 1 + Sum_{n>=1} (x^(n*(n+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^n))).

			Triangle begins:
   1;
  -1;
   1,  1;
  -1, -1;
   1,  0;
  -1,  0,  1;
   1,  1, -1;
  -1, -1,  0;
   1,  0, -1;
  -1,  0,  2,  1;
   1,  1, -1, -1;
  -1, -1,  1,  0;
   1,  0, -2, -1;
  -1,  0,  2,  1;
   1,  1, -2,  0,  1;
  -1, -1,  2,  1, -1;
   1,  0, -2, -1,  0;
  -1,  0,  3,  1, -1;
   1,  1, -3, -2,  1;
  -1, -1,  2,  1, -1;
   1,  0, -3, -1,  2,  1;
  -1,  0,  4,  2, -1, -1;
   1,  1, -3, -1,  2,  0;
  -1, -1,  3,  1, -3, -1;
   1,  0, -4, -2,  2,  1;
  -1,  0,  4,  2, -3, -1;
   1,  1, -4, -2,  3,  1;
  -1, -1,  4,  2, -3,  0,  1;
   1,  0, -4, -2,  4,  2, -1;
  -1,  0,  5,  2, -4, -2,  0;
   1,  1, -5, -2,  5,  1, -1;
  -1, -1,  4,  2, -5, -2,  1;
   1,  0, -5, -2,  5,  2, -1;
  -1,  0,  6,  3, -6, -3,  1;
   1,  1, -5, -3,  6,  2, -1;
  -1, -1,  5,  2, -7, -2,  3,  1;
  ...

Paul D. Hanna, Table of n, a(n) for n = 1..10075

Cf. A003406, A008289.

PARI
```
{T(n, k)=if(n
				
```

/* Using generating formula for columns */
{tr(n) = floor((sqrt(1+8*n)-1)/2)} \\ number of terms in row n
{T(n,k) = polcoeff( x^(k*(k+1)/2) / prod(j=1,k, 1 + x^j +x*O(x^n)), n)}
{for(n=1,50, for(k=1, tr(n), print1(T(n,k),", "));print(""))} \\ Paul D. Hanna, Jan 28 2024

G.f. of column k: x^(k*(k+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^k)) for k >= 1.

Showing 1-10 of 13 results. Next

cp's OEIS Frontend

A006140 Erroneous version of A003406.

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A158690 Expansion of the basic hypergeometric series 1 + (1 - exp(-t)) + (1 - exp(-t))*(1 - exp(-3*t)) + (1 - exp(-t))*(1 - exp(-3*t))*(1 - exp(-5*t)) + ... as a series in t.

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A117193 Number of partitions of n into distinct parts with an odd rank.

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A117192 Number of partitions of n into distinct parts with an even rank.

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A005895 Weighted count of partitions with distinct parts.

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A003475 Expansion of Sum_{k>0} (-1)^(k+1) q^(k^2) / ((1-q)(1-q^3)(1-q^5)...(1-q^(2k-1))).

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A291954 Triangle read by rows: T(n,k) = T(n-k,k-1) - T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A203568 a(n) = A026837(n) - A026838(n).

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A158690 Expansion of the basic hypergeometric series 1 + (1 - exp(-t)) + (1 - exp(-t))(1 - exp(-3t)) + (1 - exp(-t))(1 - exp(-3t))(1 - exp(-5t)) + ... as a series in t.