A010054 -id:A010054

A365800

Dirichlet inverse of the characteristic function of triangular numbers (A010054).

+20
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,18

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..101025

FORMULA

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A010054(n/d) * a(d).

PROG

(PARI)

A010054(n) = issquare(8*n + 1);

memoA365800 = Map();

A365800(n) = if(1==n, 1, my(v); if(mapisdefined(memoA365800, n, &v), v, v = -sumdiv(n, d, if(d<n, A010054(n/d)*A365800(d), 0)); mapput(memoA365800, n, v); (v)));

CROSSREFS

Cf. A000217, A010054, A365799 (parity of terms).

Absolute values differ from A308264.

KEYWORD

sign

AUTHOR

Antti Karttunen, Oct 05 2023

STATUS

approved

A109413

a(n) equals the (n*(n+1)/2)-th partial sum of the self-convolution of A010054, which has the g.f.: Sum_{n>=0} x^(n*(n+1)/2).

+20
2

1, 3, 6, 11, 17, 24, 35, 45, 58, 71, 87, 106, 123, 144, 164, 189, 216, 240, 269, 298, 329, 365, 396, 437, 471, 510, 551, 591, 642, 683, 730, 778, 827, 882, 932, 987, 1048, 1105, 1165, 1220, 1289, 1355, 1418, 1485, 1549, 1626, 1699, 1772, 1846, 1923, 2002, 2080

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..51.

FORMULA

a(n) = [x^(n*(n+1)/2)] [Sum_{k>=0} x^(k*(k+1)/2)]^2/(1-x).

PROG

(PARI) {a(n)=local(X=x+x*O(x^(n*(n+1)/2))); polcoeff((eta(X^2)^2/eta(X))^2/(1-X), n*(n+1)/2)}

CROSSREFS

Cf. A109414, A109415, A010054.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 27 2005

STATUS

approved

A109414

a(n) equals the (n*(n+1)/2)-th partial sum of the self-convolution cube of A010054, which has the g.f.: Sum_{k>=0} x^(k*(k+1)/2).

+20
2

1, 4, 11, 26, 54, 90, 151, 232, 326, 456, 612, 811, 1030, 1304, 1607, 1953, 2383, 2812, 3329, 3893, 4515, 5226, 5983, 6809, 7703, 8718, 9762, 10891, 12160, 13475, 14868, 16380, 17986, 19699, 21524, 23415, 25482, 27658, 29923, 32288, 34814, 37452

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..41.

FORMULA

a(n) = [x^(n*(n+1)/2)] [Sum_{k>=0} x^(k*(k+1)/2)]^3/(1-x).

PROG

(PARI) {a(n)=local(X=x+x*O(x^(n*(n+1)/2))); polcoeff((eta(X^2)^2/eta(X))^3/(1-X), n*(n+1)/2)}

CROSSREFS

Cf. A109413, A109415, A010054.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 27 2005

STATUS

approved

A109415

a(n) equals the (n*(n+1)/2)-th partial sum of the self-convolution 4th power of A010054, which has the g.f.: Sum_{k>=0} x^(k*(k+1)/2).

+20
2

1, 5, 19, 58, 152, 309, 585, 1046, 1666, 2601, 3871, 5508, 7680, 10423, 13835, 17984, 23168, 29225, 36431, 45000, 54780, 66299, 79637, 94546, 111612, 131215, 152683, 177008, 204264, 234375, 267795, 304678, 345240, 389213, 438235, 490842, 548542

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..36.

FORMULA

a(n) = [x^(n*(n+1)/2)] [Sum_{k>=0} x^(k*(k+1)/2)]^4/(1-x).

PROG

(PARI) {a(n)=local(X=x+x*O(x^(n*(n+1)/2))); polcoeff((eta(X^2)^2/eta(X))^4/(1-X), n*(n+1)/2)}

CROSSREFS

Cf. A109413, A109414, A010054.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 27 2005

STATUS

approved

A181649

An INVERT sequence for A010054.

+20
2

1, 1, 2, 3, 6, 10, 18, 32, 57, 101, 179, 319, 566, 1006, 1786, 3174, 5638, 10016, 17793, 31609, 56153, 99753, 177211, 314810, 559255, 993501, 1764935, 3135366, 5569909, 9894819, 17577926, 31226796, 55473705, 98547807, 175067983, 311004383

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Eigensequence of the sequence array for A010054.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: 1/(1-x*Product{k>0,(1 - x^(2k))/(1-x^(2k-1))}).

G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 + x^1 / (1 - x / (1 + x / (1 + x^2 / (1 - x / (1 + x / (1 + x^3 / (1 - x / (1 + x / ...)))))))))))). - Michael Somos, Jan 03 2013

a(n) ~ c / r^n, where r = 0.5629116358141452127351993944163442032777187438473224785071475357915... is the root of the equation (-1 + x)*QPochhammer(x^2, x^2) = QPochhammer(1/x, x^2) and c = 0.5730261147067572839709085685318242468812339379480160560847761872213851... - Vaclav Kotesovec, Jan 23 2024

EXAMPLE

1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 18*x^6 + 32*x^7 + 57*x^8 + 101*x^9 + ...

MATHEMATICA

eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[1/(1 - q^(7/8)*eta[q^2]^2/eta[q]), {q, 0, 50}], q] (* G. C. Greubel, Sep 16 2018 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / (1 - x * eta(x^2 + A)^2 / eta(x + A)), n))} /* Michael Somos, Jan 03 2013 */

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Nov 03 2010

STATUS

approved

A365799

Parity of the Dirichlet inverse of the characteristic function of triangular numbers (A010054).

+20
2

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..101025

Index entries for characteristic functions

FORMULA

a(n) = A000035(A365800(n)).

PROG

(PARI) A365799(n) = (A365800(n)%2); \\ Uses also code from A365800.

CROSSREFS

Sequence A365800 reduced modulo 2.

Cf. A000035, A000217, A010054.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 05 2023

STATUS

approved

A093523

Inverse binomial transform of A010054 (1 if triangular number else 0).

+20
1

1, 0, -1, 3, -7, 14, -24, 34, -35, 8, 82, -298, 759, -1704, 3627, -7538, 15425, -30992, 60673, -114647, 206853, -351365, 549132, -752653, 784277, -162126, -2252600, 8950526, -25129652, 61349528, -138789534, 299803944, -629297799, 1298075184, -2650139349, 5375982063, -10849417306

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

The e.g.f., F(x) = exp(-x)*sum_{n>=0} x^(n*(n+1)/2)/(n*(n+1)/2)!, is approximated by 1/sqrt(2x) for x>1; example: F(1)=0.79758, F(2)=0.59852, F(10)=0.23183, F(50)=0.10063.

LINKS

Table of n, a(n) for n=0..36.

FORMULA

E.g.f.: exp(-x)*sum_{n>=0} x^(n*(n+1)/2)/(n*(n+1)/2)!

MATHEMATICA

Table[Sum[(-1)^(n-k) * Binomial[n, k] * SquaresR[1, 8*k+1]/2, {k, 0, n}], {n, 0, 40}] (* Vaclav Kotesovec, Oct 31 2017 *)

PROG

(PARI) {a(n)=n!*polcoeff((sum(k=0, sqrtint(2*n+1), x^(k*(k+1)/2)/(k*(k+1)/2)!)*sum(j=0, n, (-x)^j/j!)+x*O(x^n)), n)}

KEYWORD

sign

AUTHOR

Paul D. Hanna, Mar 30 2004

STATUS

approved

A143961

Binomial transform of A010054 (characteristic function of triangular numbers).

+20
1

1, 2, 3, 5, 9, 16, 28, 50, 93, 178, 342, 650, 1223, 2302, 4383, 8480, 16609, 32658, 63973, 124375, 240181, 462597, 893388, 1737375, 3407429, 6729596, 13336532, 26415118, 52129356, 102359648, 200067986, 389989828, 760206585, 1485887074, 2917775247, 5759836217

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Transforms

FORMULA

a(n) = Sum_{k=0..n} C(n,k) * A010054(k).

EXAMPLE

a(4) = [1,4,6,4,1]*[1,1,0,1,0] = 1+4+4 = 9.

MAPLE

a:= proc(n) local k, i, s; k:=0; i:=0; s:=0; while k<=n do s:= s+binomial(n, k); i:=i+1; k:=k+i; od; s end: seq(a(n), n=0..40);

MATHEMATICA

Table[Sum[Binomial[n, k] * SquaresR[1, 8*k+1]/2, {k, 0, n}], {n, 0, 40}] (* Vaclav Kotesovec, Oct 30 2017 *)

CROSSREFS

Cf. A000217, A007318, A010054.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 05 2008

STATUS

approved

A167618

Convolution of A010054 with A052343.

+20
1

1, 2, 2, 3, 3, 2, 4, 5, 2, 4, 5, 3, 6, 5, 3, 4, 8, 5, 4, 6, 2, 9, 8, 4, 6, 6, 5, 7, 8, 3, 8, 11, 6, 4, 8, 5, 7, 12, 5, 9, 7, 5, 10, 8, 6, 8, 12, 5, 9, 12, 3, 10, 14, 3, 6, 8, 10, 14, 11, 8, 6, 14, 5, 7, 9, 8, 15, 14, 5, 6, 14, 8, 13, 11, 6, 9, 15, 8, 8, 15, 5, 12, 17, 6, 15, 8, 8, 16, 9, 6, 7, 19, 8, 15

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n) = SUM(A010054(k)*A052343(n-k): 0<=k<=n);

A002636(n) <= a(n).

LINKS

Table of n, a(n) for n=0..93.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Nov 07 2009

STATUS

approved

A320784

Negated inverse Euler transform of {-1 if n is a triangular number else 0, n > 0} = -A010054.

+20
1

1, 1, 0, 1, 1, 1, 2, 3, 3, 5, 8, 11, 14, 23, 31, 47, 68, 101, 144, 217, 315, 471, 693, 1035, 1528, 2287, 3397, 5085, 7587, 11377, 17017, 25565, 38349, 57681, 86724, 130645, 196778, 296853, 447864, 676479, 1022082, 1545685, 2338299, 3540111, 5361606, 8125551

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

COMMENTS

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

LINKS

Table of n, a(n) for n=0..45.

OEIS Wiki, Euler transform

MATHEMATICA

EulerInvTransform[{}]={}; EulerInvTransform[seq_]:=Module[{final={}}, For[i=1, i<=Length[seq], i++, AppendTo[final, i*seq[[i]]-Sum[final[[d]]*seq[[i-d]], {d, i-1}]]];

Table[Sum[MoebiusMu[i/d]*final[[d]], {d, Divisors[i]}]/i, {i, Length[seq]}]];

-EulerInvTransform[-Table[SquaresR[1, 8*n+1]/2, {n, 30}]]

CROSSREFS

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.

Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.

Inverse Euler transforms: A059966, A320767, A320776, A320777, A320778, A320779, A320780, A320781, A320782.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 22 2018

STATUS

approved