A375106 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+3)).

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

Offset

0,7

Links

Table of n, a(n) for n=0..95.

R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293. see p. 286.

Formula

G.f.: Product_{k>0} (1-x^(7*k))^2 / ((1-x^(7*k-1)) * (1-x^(7*k-6))).

G.f.: Sum_{k in Z} x^(3*k) / (1 - x^(7*k+1)).

Prog

(PARI) my(N=100, x='x+O('x^N)); Vec(sum(k=-N, N, x^k/(1-x^(7*k+3))))
(PARI) my(N=100, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^2/((1-x^(7*k-1))*(1-x^(7*k-6)))))

Crossrefs

Cf. A374900, A375148, A375149, A375150.

Cf. A375107, A375108.

Keyword

nonn,new

Author

Seiichi Manyama, Jul 30 2024

Status

approved

A318451 The 2-adic valuation of A318450.

0, 0, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 2, 0, 1, 3, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 1, 2, 1, 0, 2, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 3, 3, 2, 1, 1, 4, 2, 1, 2, 1, 1, 2, 1, 1, 4, 0, 2, 2, 1, 1, 2, 2, 1, 3, 1, 1, 4, 1, 2, 2, 1, 1, 7, 1, 1, 2, 2, 1, 2, 1, 1, 4, 2, 1, 2, 1, 2, 1, 1, 3, 4, 3, 1, 2, 1, 1, 3

Offset

1,9

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = A007814(A318450(n)).

Prog

(PARI) A318451(n) = valuation(A318450(n), 2); \\ Needs also code from A318450.

Crossrefs

Cf. A007814, A318450.

Cf. also A046645, A317946, A318315, A318455.

Keyword

nonn

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Status

approved

A078805 Triangular array T given by T(n,k)= number of 01-words of length n having exactly k 1's, every runlength of 1's odd and initial letter 0.

1, 1, 1, 1, 2, 0, 1, 3, 1, 1, 1, 4, 3, 2, 0, 1, 5, 6, 4, 2, 1, 1, 6, 10, 8, 6, 2, 0, 1, 7, 15, 15, 13, 6, 3, 1, 1, 8, 21, 26, 25, 16, 9, 2, 0, 1, 9, 28, 42, 45, 36, 22, 9, 4, 1, 1, 10, 36, 64, 77, 72, 50, 28, 12, 2, 0, 1, 11, 45, 93, 126, 133, 106, 70, 34, 13, 5, 1, 1, 12, 55, 130, 198, 232

Offset

1,5

Comments

Row sums: A028495.

References

Clark Kimberling, Binary words with restricted repetitions and associated compositions of integers, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 141-151.

Links

Table of n, a(n) for n=1..84.

Formula

T(n, k)=T(n-2, k)+T(n-2, k-1)+T(n-2, k-2)+T(n-3, k-1)-T(n-4, k-2) for 0<=k<=n, n>=1. (All numbers T(i, j) not in the array are 0, by definition of T.)

Example

T(5,2) counts the words 01010, 01001, 00101. Top of triangle T:
1 = T(1,0)
1 1 = T(2,0) T(2,1)
1 2 0 = T(3,0) T(3,1) T(3,2)
1 3 1 1
1 4 3 2 0

Crossrefs

Cf. A078804, A078806.

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 07 2002

Status

approved

A065432 Triangle related to Catalan triangle: recurrence related to A033877 (Schroeder numbers).

1, 1, -1, 1, -2, 0, 1, -3, 1, 1, 1, -4, 3, 2, 0, 1, -5, 6, 2, -2, -2, 1, -6, 10, 0, -6, -4, 0, 1, -7, 15, -5, -11, -3, 5, 5, 1, -8, 21, -14, -15, 4, 15, 10, 0, 1, -9, 28, -28, -15, 19, 26, 6, -14, -14, 1, -10, 36, -48, -7, 42, 30, -16, -42, -28, 0, 1, -11, 45, -75, 14, 70, 16, -60, -70, -14, 42, 42, 1, -12, 55, -110, 54, 96, -28, -120, -70, 56, 126, 84, 0

Offset

0,5

Comments

Sums of odd rows are 0, of even rows are the Catalan numbers (A000108) with alternating signs. Row sums of unsigned version give A065441.

Links

Table of n, a(n) for n=0..90.

Example

Triangle starts:
[0] 1;
[1] 1, -1;
[2] 1, -2,  0;
[3] 1, -3,  1,   1;
[4] 1, -4,  3,   2,   0;
[5] 1, -5,  6,   2,  -2, -2;
[6] 1, -6, 10,   0,  -6, -4,  0;
[7] 1, -7, 15,  -5, -11, -3,  5,  5;
[8] 1, -8, 21, -14, -15,  4, 15, 10,   0;
[9] 1, -9, 28, -28, -15, 19, 26,  6, -14, -14.

Mathematica

a[0, 0] := 1; a[n_, k_] := 0/; (k > n||n < 0||k < 0); a[n_, k_] := a[n, k] = a[n, k-1]-2a[n-1, k-1]+a[n-1, k]; Table[a[n, k], {n, 0, 16}, {k, 0, n}]

Keyword

sign,tabl

Author

Wouter Meeussen, Nov 16 2001

Extensions

More terms from Sean A. Irvine, Sep 03 2023

Status

approved

A329801 Expansion of Sum_{k>=1} x^(k*(k + 1)/2) / (1 + x^(k*(k + 1)/2)).

1, -1, 2, -1, 1, -1, 1, -1, 2, 0, 1, -3, 1, -1, 3, -1, 1, -1, 1, -2, 3, -1, 1, -3, 1, -1, 2, 0, 1, -1, 1, -1, 2, -1, 1, -2, 1, -1, 2, -2, 1, -2, 1, -1, 4, -1, 1, -3, 1, 0, 2, -1, 1, -1, 2, -2, 2, -1, 1, -5, 1, -1, 3, -1, 1, 0, 1, -1, 2, 0, 1, -4, 1, -1, 3, -1, 1, 0, 1, -2, 2, -1, 1, -3, 1

Offset

1,3

Links

Table of n, a(n) for n=1..85.

Formula

G.f.: Sum_{k>=1} (-1)^(k + 1) * theta_2(x^(k/2)) / (2 * x^(k/8)).

a(n) = Sum_{d|n} (-1)^(n/d + 1) * A010054(d).

Mathematica

nmax = 85; CoefficientList[Series[Sum[x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^(n/d + 1) Boole[IntegerQ[Sqrt[8 d + 1]]], {d, Divisors[n]}], {n, 1, 85}]

Crossrefs

Cf. A000217, A007862, A010054, A048272, A304876, A317529.

Keyword

sign

Author

Ilya Gutkovskiy, Nov 21 2019

Status

approved

A094184 Triangle read by rows in which each term equals the entry above minus the entry left plus twice the entry left-above.

1, 1, 1, 1, 2, 0, 1, 3, 1, -1, 1, 4, 3, -2, 0, 1, 5, 6, -2, -2, 2, 1, 6, 10, 0, -6, 4, 0, 1, 7, 15, 5, -11, 3, 5, -5, 1, 8, 21, 14, -15, -4, 15, -10, 0, 1, 9, 28, 28, -15, -19, 26, -6, -14, 14, 1, 10, 36, 48, -7, -42, 30, 16, -42, 28, 0, 1, 11, 45, 75, 14, -70, 16, 60, -70, 14, 42, -42, 1, 12, 55, 110, 54, -96, -28, 120, -70, -56, 126, -84, 0, 1

Offset

0,5

Comments

Row sums are A086990 or A090412. (Superseeker finds that the j-th coefficient of OGF(A090412)(z)*(1-z)^j equals A049122). Same absolute values as A065432. Even rows end in 0, odd rows end in Catalan numbers (A000118) with alternating sign.

Links

Table of n, a(n) for n=0..91.

Formula

T(i, j)=T(i-1, j)-T(i, j-1)+2*T(i-1, j-1), with T(i, 0)=1 and T(i, j)=0 if j>i.

Example

Table starts {1},{1,1},{1,2,0},{1,3,1,-1},{1,4,3,-2,0},{1,5,6,-2,-2,2}

Mathematica

T[_, 0]:=1; T[0, 0]:=1; T[i_, j_]/; j>i:=0; T[i_, j_]:=T[i, j]=T[i-1, j]-T[i, j-1]+2 T[i-1, j-1]

Crossrefs

Cf. A086990, A090412, A049122, A009766, A065432, A065441.

Keyword

sign,tabl

Author

Wouter Meeussen, May 06 2004

Status

approved