A025192 -id:A025192 - cp's OEIS frontend

A347237 Möbius transform of A347236.

1, 0, 1, 6, 1, 0, 3, 6, 17, 0, 1, 6, 3, 0, 1, 42, 1, 0, 3, 6, 3, 0, 5, 6, 37, 0, 49, 18, 1, 0, 5, 78, 1, 0, 3, 102, 3, 0, 3, 6, 1, 0, 3, 6, 17, 0, 5, 42, 89, 0, 1, 18, 5, 0, 1, 18, 3, 0, 1, 6, 5, 0, 51, 330, 3, 0, 3, 6, 5, 0, 1, 102, 5, 0, 37, 18, 3, 0, 3, 42, 353, 0, 5, 18, 1, 0, 1, 6, 7, 0, 9, 30, 5, 0, 3, 78, 3, 0, 17

Offset: 1

Antti Karttunen, Aug 24 2021

Dirichlet convolution of A003972 (prime shifted phi) with A061019.
Dirichlet convolution of A003961 with A158523.
Multiplicative because A003972 and A061019 are, and also because A347236 is.
From Antti Karttunen, Aug 25 2021: (Start)
All terms are nonnegative because sequence is multiplicative and a(p^k) >= 0 for all primes p and k >= 0.
Proof: For any prime p, sequence a(p^k), k>=0, is obtained as an ordinary convolution of sequences (-p)^k and the first differences of q^k, where q = A151800(p). (E.g., for powers of 2, the sequences convolved are A122803 and A025192, giving A102901.) This convolution is an alternating sum, with the terms 1*(q-1)*q^(k-1), -(p)*(q-1)*q^(k-2), (p^2)*(q-1)*q^(k-3), -(p^3)*(q-1)*q^(k-4), ..., (p^(k-1))*(q-1), -(p^k), for odd k, with sum of each consecutive pair being nonnegative because q >= p+1, while with an even exponent k, the leftover term p^k at the end is also positive, thus the whole sum is nonnegative also in that case.
(End)

Cf. A000040, A001223, A003961, A003972, A008683, A008836, A016825 (positions of zeros), A061019, A102901, A120612, A158523, A347236.
Cf. also A347137.
Cf. A151800.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347237(n) = sumdiv(n,d,A061019(d)*eulerphi(A003961(n/d)));
\\ Or alternatively as:
A158523(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); ((-1)^e)*(p+1)*(p^(e-1))); };
A347237(n) = sumdiv(n,d,A003961(n/d)*A158523(d));

a(n) = Sum_{d|n} A008683(n/d) * A347236(d).
a(n) = Sum_{d|n} A003972(n/d) * A061019(d).
a(n) = Sum_{d|n} A003961(n/d) * A158523(d).
For all n >= 1, a(A000040(n)) = A001223(n) - 1.
For all n >= 0, a(2^n) = A102901(n).
For all n >= 0, a(3^n) = A120612(n).
Multiplicative with a(p^e) = (-p)^e + (A151800(p)-1)*(A151800(p)^e-(-p)^e)/(A151800(p)+p). - Sebastian Karlsson, Sep 02 2021

A364825 G.f. satisfies A(x) = 1 - xA(x)^3 (1 - 3*A(x)).

Original entry on oeis.org

1, 2, 18, 222, 3166, 49098, 804138, 13686198, 239671590, 4290463698, 78160665666, 1444298971662, 27005948771886, 510024567278234, 9714561608833242, 186403770207998310, 3599812021110287862, 69914211761486437026, 1364692279095996581490

Offset: 0

Seiichi Manyama, Aug 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..757

Cf. A025192, A107841, A235347, A364826, A364827.

Cf. A144097, A243659.

A364825 := proc(n)
    (-1)^n*add( (-3)^k*binomial(n,k) * binomial(3*n+k+1,n)/(3*n+k+1),k=0..n) ;
end proc:
seq(A364825(n),n=0..80); # R. J. Mathar, Aug 10 2023

a(n) = (-1)^n*sum(k=0, n, (-3)^k*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1));

a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(3*n+k+1,n) / (3*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(3*n,k-1) for n > 0.
D-finite with recurrence +2079*n*(3*n-1)*(3*n+1)*a(n) +(-347173*n^3 +395007*n^2 -41030*n -43092)*a(n-1) +18*(-59207*n^3 +325826*n^2 -590255*n +352406)*a(n-2) +3*(-3299*n^3 +35998*n^2 -125399*n +141144)*a(n-3) +9*(3*n-10)*(3*n-11) *(n-4)*a(n-4)=0. - R. J. Mathar, Aug 10 2023

A364826 G.f. satisfies A(x) = 1 - xA(x)^4 (1 - 3*A(x)).

Original entry on oeis.org

1, 2, 22, 338, 6038, 117570, 2420758, 51833106, 1142472150, 25749801986, 590737764118, 13748997055826, 323842714201622, 7704914865207362, 184899022770465558, 4470200057557410834, 108776308617293352534, 2662072268791363675650

Offset: 0

Seiichi Manyama, Aug 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..704

Cf. A025192, A107841, A235347, A364825, A364827.

Cf. A243667, A260332.

a(n) = (-1)^n*sum(k=0, n, (-3)^k*binomial(n, k)*binomial(4*n+k+1, n)/(4*n+k+1));

a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(4*n+k+1,n) / (4*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(5*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(4*n,k-1) for n > 0.

A364827 G.f. satisfies A(x) = 1 - xA(x)^5 (1 - 3*A(x)).

Original entry on oeis.org

1, 2, 26, 478, 10254, 240122, 5950530, 153417542, 4072868742, 110585691634, 3056671795946, 85722961493742, 2433127206219582, 69763483031049066, 2017643094336224914, 58789801741123032918, 1724199860717303739062, 50858327392484088101346

Offset: 0

Seiichi Manyama, Aug 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..666

Cf. A025192, A107841, A235347, A364825, A364826.

Cf. A243668, A363006.

a(n) = (-1)^n*sum(k=0, n, (-3)^k*binomial(n, k)*binomial(5*n+k+1, n)/(5*n+k+1));

a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(5*n+k+1,n) / (5*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(6*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(5*n,k-1) for n > 0.

A378194 Rectangular array, read by descending antidiagonals: row n shows the integers m such that the number of primes of the form 4k+3 (including multiplicities) that divide m is n-1.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 7, 18, 27, 8, 11, 21, 54, 81, 10, 12, 33, 63, 162, 243, 13, 14, 36, 99, 189, 486, 729, 16, 15, 42, 108, 297, 567, 1458, 2187, 17, 19, 45, 126, 324, 891, 1701, 4374, 6561, 20, 22, 49, 135, 378, 972, 2673, 5103, 13122, 19683, 25, 23, 57, 147, 405, 1134, 2916, 8019, 15309, 39366, 59049, 26, 24, 66, 171, 441, 1215, 3402, 8748, 24057, 45927, 118098, 177147

Offset: 1

Clark Kimberling, Jan 14 2025

Every positive integer occurs exactly once.

			Corner:
      1     2     4     5     8     10     13     16      17
      3     6     7    11    12     14     15     19      22
      9    18    21    33    36     42     45     49      57
     27    54    63    99   108    126    135    147     171
     81   162   189   297   324    378    405    441     513
    243   486   567   891   972   1134   1215   1323    1539
    729  1458  1701  2673  2916   3402   3645   3969    4617
   2187  4374  5103  8019  8748  10206  10935  11907   13851

Cf. A065339, A002144, A002145, A376961, A378193, A072437 (row 1), A000244 (column 0), A025192 (column 1).

A378194 := proc(n, k)
    option remember;
    local a;
    if k = 0 then
        0;
    else
        for a from procname(n, k-1)+1 do
            if A065339(a) = n-1 then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(seq( A378194(n, d-n), n=1..d-1), d=2..10) ; # R. J. Mathar, Jan 28 2025

u = Map[Map[#[[1]] &, #] &, GatherBy[
    SortBy[Map[{#, 1 + Count[Map[IntegerQ[(# - 3)/4] && PrimeQ[#] &,
             Flatten[Map[ConstantArray[#[[1]], #[[2]]] &,
             FactorInteger[#]]]], True]} &,
      Range[24000]], #[[2]] &], #[[2]] &]];
r[m_] := Take[u[[m]], 10];
w[m_, n_] := r[m][[n]];
Grid[Table[w[m, n], {m, 1, 8}, {n, 1, 9}]]   (* array *)
Table[w[n - k + 1, k], {n, 8}, {k, n, 1, -1}] // Flatten  (* sequence *)
(* Peter J. C. Moses, Nov 19 2024 *)

Definition corrected. - R. J. Mathar, Jan 28 2025

A062177 Shifts left when MASKCONVolved with itself.

Original entry on oeis.org

1, 1, 2, 4, 12, 24, 72, 192, 720, 1440, 4320, 11520, 43200, 103680, 362880, 1105920, 4665600, 9331200, 27993600, 74649600, 279936000, 671846400, 2351462400, 7166361600, 30233088000, 67184640000, 221709312000, 644972544000

Offset: 0

Antti Karttunen, Jun 12 2001

Note that the factorials 1!, 2!, 4!, 6!, 9! can be found from the positions 1,3,6,9,15 (or 2,4,7,10,16 if zero-based indexing is used) of this sequence. I do not know whether any larger factorials occur in the sequence.

N. J. A. Sloane, Transforms

Other self-convolved sequences: A000108, A007460 - A007464, A025192, A038044, A061922.

EIGENbyMASKCONV := proc(upto_n) local n,a,j,i,s,m; a := [1]; for i from 0 to upto_n do s := 0; m := maskees(i); n := nops(m); for j from 1 to n do s := s+(a[m[j]+1]*a[m[(n-j)+1]+1]); od; a := [op(a),s]; od; RETURN(a); end;
maskees := proc(n) local a,b,u,i; a := []; b := list_mask_bits(n); u := (2^nops(b))-1; for i from 0 to u do a := [op(a),sum_by_mask_list(i,b)]; od; RETURN(a); end;
list_mask_bits := proc(nn) local n,a,x; n := nn; x := 1; a := []; while(n > 0) do if(1 = (n mod 2)) then a := [op(a),x]; fi; n := floor(n/2); x := 2*x; od; RETURN(a); end;
sum_by_mask_list := proc(nn,a) local n,i,s; n := nn; s := 0; i := 1; while(n > 0) do if(1 = (n mod 2)) then s := s + a[i]; fi; n := floor(n/2); i := i+1; od; RETURN(s); end;

A067337 Triangle where T(n,k)=2*T(n,k-1)+C(n-1,k)-C(n-1,k-1) and n>=k>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 5, 9, 1, 4, 8, 14, 27, 1, 5, 12, 22, 41, 81, 1, 6, 17, 34, 63, 122, 243, 1, 7, 23, 51, 97, 185, 365, 729, 1, 8, 30, 74, 148, 282, 550, 1094, 2187, 1, 9, 38, 104, 222, 430, 832, 1644, 3281, 6561, 1, 10, 47, 142, 326, 652, 1262, 2476, 4925, 9842

Offset: 0

Henry Bottomley, Jan 15 2002

			Rows start 1; 1,1; 1,2,3; 1,3,5,9; 1,4,8,14,27; etc. T(4,0)=2*0+1-0=1; T(4,1)=2*1+3-1=4; T(4,2)=2*4+3-3=8; T(4,3)=2*8+1-3=14; T(4,4)=2*14+0-1=27.

Row sums are A025192. Columns include A000012, A000027 and A022856 (essentially). Right hand columns include A000244 (essentially), A007051 and A047926. Central diagonal is A067336.

T(n, k)=2*T(n, k-1)+A037012(n, k). T(n, k)=T(n-1, k-1)+T(n-1, k) if k0.

A081954 Triangle read by rows: T(n, k) = 2^(n-k)*3^k, n >= 1, 0 <= k < n.

Original entry on oeis.org

2, 4, 6, 8, 12, 18, 16, 24, 36, 54, 32, 48, 72, 108, 162, 64, 96, 144, 216, 324, 486, 128, 192, 288, 432, 648, 972, 1458, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 1024, 1536, 2304, 3456, 5184, 7776

Offset: 1

Amarnath Murthy, Apr 02 2003

			Triangle begins:
2
4 6
8 12 18
16 24 36 54
32 48 72 108 162
...

Leading diagonal is given by A025192.

Cf. A025192, A081955, A081956, A002262, A090404.

tabl(nrows) = {for (n=1, nrows, for (k=0, n-1, print1(2^(n-k)*3^k, ", ");); print(););} \\ Michel Marcus, May 14 2013

Corrected and extended by David Wasserman, Jul 21 2004

A094039 Binomial transform of (Jacobsthal(n) + 2^n*Jacobsthal(-n))/2.

Original entry on oeis.org

0, 1, 2, 6, 16, 46, 132, 386, 1136, 3366, 10012, 29866, 89256, 267086, 799892, 2396946, 7185376, 21545206, 64613772, 193797626, 581305496, 1743741726, 5230875652, 15691927906, 47074385616, 141220360646, 423655489532

Offset: 0

Paul Barry, Apr 23 2004

Binomial transform of Jacobsthal(n)(1-(-1)^n)/2.

Starting with "1" = A001045 convolved with A025192: i.e., (1, 2, 6, 16, 46, ...) = (1, 1, 3, 5, 11, ...) * (1, 2, 6, 18, 54, ...). - Gary W. Adamson, May 10 2013

Index entries for linear recurrences with constant coefficients, signature (4,-1,-6).

Cf. A001045, A094038, A051450.

Cf. A025192.

G.f.: x(1-2x-x^2)/((1+x)(1-2x)(1-3x))

a(n) = 3^n/6 - (-1)^n/6 + 2^n/6 - 0^n/6.

A114188 Riordan array (1/(1-x),x(1+x)/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 7, 1, 1, 16, 26, 10, 1, 1, 25, 70, 52, 13, 1, 1, 36, 155, 190, 87, 16, 1, 1, 49, 301, 553, 403, 131, 19, 1, 1, 64, 532, 1372, 1462, 736, 184, 22, 1, 1, 81, 876, 3024, 4446, 3206, 1216, 246, 25, 1, 1, 100, 1365, 6084, 11826, 11584, 6190, 1870, 317

Offset: 0

Paul Barry, Nov 16 2005

Product of A007318 and A113953, that is, (1/(1-x),x/(1-x))*(1,x(1+2x)).
Row sums are A025192. Diagonal sums are A052980.
Inverse is A114189. A signed version is A110511.

			Triangle begins
1;
1, 1;
1, 4, 1;
1, 9, 7, 1;
1, 16, 26, 10, 1;
1, 25, 70, 52, 13, 1;
1, 36,155,190, 87, 16, 1;

P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.

Cf. A007318, A025192, A052980, A110511, A113953, A114189.

T(n, k) = Sum_{j=0..n} C(n, j)*C(k, j-k)2^(j-k).
T(n, k) = Sum_{j=0..n-k} C(k, j)*C(n+k-j, 2k).
T(n,k) = 2*T(n-1,k)+T(n-1,k-1)-T(n-2,k)+T(n-2,k-1), T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 11 2014
G.f.: 1/(1-y-x*(1+y)/(1-y)). - Vladimir Kruchinin, Apr 21 2015

cp's OEIS Frontend

A347237 Möbius transform of A347236.

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A364825 G.f. satisfies A(x) = 1 - x*A(x)^3 * (1 - 3*A(x)).

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A364826 G.f. satisfies A(x) = 1 - x*A(x)^4 * (1 - 3*A(x)).

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A364827 G.f. satisfies A(x) = 1 - x*A(x)^5 * (1 - 3*A(x)).

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A378194 Rectangular array, read by descending antidiagonals: row n shows the integers m such that the number of primes of the form 4k+3 (including multiplicities) that divide m is n-1.

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A062177 Shifts left when MASKCONVolved with itself.

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A067337 Triangle where T(n,k)=2*T(n,k-1)+C(n-1,k)-C(n-1,k-1) and n>=k>=0.

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A081954 Triangle read by rows: T(n, k) = 2^(n-k)*3^k, n >= 1, 0 <= k < n.

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A094039 Binomial transform of (Jacobsthal(n) + 2^n*Jacobsthal(-n))/2.

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A364825 G.f. satisfies A(x) = 1 - xA(x)^3 (1 - 3*A(x)).

A364826 G.f. satisfies A(x) = 1 - xA(x)^4 (1 - 3*A(x)).

A364827 G.f. satisfies A(x) = 1 - xA(x)^5 (1 - 3*A(x)).