A144074 -id:A144074 - cp's OEIS frontend

A034899 Euler transform of powers of 2 [ 2,4,8,16,... ].

1, 2, 7, 20, 59, 162, 449, 1200, 3194, 8348, 21646, 55480, 141152, 356056, 892284, 2221208, 5497945, 13533858, 33151571, 80826748, 196219393, 474425518, 1142758067, 2742784304, 6561052331, 15645062126, 37194451937, 88174252924, 208463595471, 491585775018

Offset: 0

N. J. A. Sloane

nonn

			From _Geoffrey Critzer_, Mar 07 2012: (Start)
Per comment in A102866, a(n) is also the number of multisets of binary words of total length n.
a(2) = 7 because the multisets are {a,a}, {b,b}, {a,b}, {aa}, {ab}, {ba}, {bb};
a(3) = 20 because the multisets are {a,a,a}, {b,b,b}, {a,a,b}, {a,b,b}, {a,aa}, {a,ab}, {a,ba}, {a,bb}, {b,aa}, {b,ab}, {b,ba}, {b,bb}, {aaa}, {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {bbb};
where the words within each multiset are separated by commas. (End)

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..3150 (first 900 terms from Alois P. Heinz)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
G. S. Venkatesh and Kurusch Ebrahimi-Fard, A Formal Power Series Approach to Multiplicative Dynamic Feedback, arXiv:2301.04949 [math.OC], 2023.
Thomas Wieder, Additional comments on this sequence

Cf. A034691, the Euler transform of 1, 2, 4, 8, 16, 32, 64, ...
Cf. A075729, A102866.
Column k=2 of A144074.
Row sums of A055375 and of A209406.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^k)^(2^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018 ~

series(1/product((1-x^(n))^(2^(n)),n=1..20),x=0,12); (Wieder)
# second Maple program:
with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add(add(d*2^d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 02 2011

nn = 20; p = Product[1/(1 - x^i)^(2^i), {i, 1, nn}]; CoefficientList[Series[p, {x, 0, nn}], x] (* Geoffrey Critzer, Mar 07 2012 *)

m=50; x='x+O('x^m); Vec(prod(k=1,m,1/(1-x^k)^(2^k))) \\ G. C. Greubel, Nov 09 2018

G.f.: 1/Product_{n>0} (1-x^n)^(2^n). - Thomas Wieder, Mar 06 2005
a(n) ~ c^2 * 2^(n-1) * exp(2*sqrt(n) - 1/2) / (sqrt(Pi) * n^(3/4)), where c = A247003 = exp( Sum_{k>=2} 1/(k*(2^k-2)) ) = 1.3976490050836502... . - Vaclav Kotesovec, Mar 09 2015
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - 2*x^k))). - Ilya Gutkovskiy, Nov 09 2018

More terms from Thomas Wieder, Mar 06 2005

A257740 Number T(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet such that all k letters occur at least once in the multiset; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 3, 14, 13, 0, 5, 49, 114, 73, 0, 7, 148, 672, 1028, 501, 0, 11, 427, 3334, 9182, 10310, 4051, 0, 15, 1170, 15030, 66584, 129485, 114402, 37633, 0, 22, 3150, 63978, 428653, 1285815, 1918083, 1394414, 394353, 0, 30, 8288, 261880, 2557972, 11117600, 24917060, 30044014, 18536744, 4596553

Offset: 0

Alois P. Heinz, May 06 2015

Row n is the inverse binomial transform of the n-th row of array A144074, which has the Euler transform of the powers of k in column k.

			T(2,2) = 3: {ab}, {ba}, {a,b}.
T(3,2) = 14: {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {a,ab}, {a,ba}, {a,bb}, {aa,b}, {ab,b}, {b,ba}, {a,a,b}, {a,b,b}.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,    3;
  0,  3,   14,    13;
  0,  5,   49,   114,     73;
  0,  7,  148,   672,   1028,     501;
  0, 11,  427,  3334,   9182,   10310,    4051;
  0, 15, 1170, 15030,  66584,  129485,  114402,   37633;
  0, 22, 3150, 63978, 428653, 1285815, 1918083, 1394414, 394353;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A000041 (for n>0), A261043, A320213, A320214, A320215, A320216, A320217, A320218, A320219, A320220.
Row sums give A257741.
Main diagonal gives A000262.
T(2n,n) gives A257742.
Cf. A144074, A319501.

A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
      d*k^d, d=numtheory[divisors](j)) *A(n-j, k), j=1..n)/n)
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*k^#&]*A[n - j, k], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 23 2017, adapted from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A144074(n,k-i).

Name changed by Alois P. Heinz, Sep 21 2018

A292804 Number A(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 5, 2, 0, 1, 4, 12, 16, 2, 0, 1, 5, 22, 55, 42, 3, 0, 1, 6, 35, 132, 225, 116, 4, 0, 1, 7, 51, 260, 729, 927, 310, 5, 0, 1, 8, 70, 452, 1805, 4000, 3729, 816, 6, 0, 1, 9, 92, 721, 3777, 12376, 21488, 14787, 2121, 8, 0

Offset: 0

Alois P. Heinz, Sep 23 2017

			A(2,2) = 5: {aa}, {ab}, {ba}, {bb}, {a,b}.
Square array A(n,k) begins:
  1, 1,   1,     1,      1,      1,       1,       1, ...
  0, 1,   2,     3,      4,      5,       6,       7, ...
  0, 1,   5,    12,     22,     35,      51,      70, ...
  0, 2,  16,    55,    132,    260,     452,     721, ...
  0, 2,  42,   225,    729,   1805,    3777,    7042, ...
  0, 3, 116,   927,   4000,  12376,   31074,   67592, ...
  0, 4, 310,  3729,  21488,  83175,  250735,  636517, ...
  0, 5, 816, 14787, 113760, 550775, 1993176, 5904746, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A000009, A102866, A256142, A292838, A292839, A292840, A292841, A292842, A292843, A292844.
Rows n=0-2 give: A000012, A001477, A000326.
Main diagonal gives A292805.
Cf. A144074, A292795, A319501.

h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
    end:
A:= (n, k)-> h(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

h[n_, i_, k_] := h[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[h[n-i*j, i-1, k]* Binomial[k^i, j], {j, 0, n/i}]]];
A[n_, k_] := h[n, n, k];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)

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

A(n,k) = Sum_{i=0..k} C(k,i) * A319501(n,i).

A256142 G.f.: Product_{j>=1} (1+x^j)^(3^j).

Original entry on oeis.org

1, 3, 12, 55, 225, 927, 3729, 14787, 57888, 224220, 860022, 3270744, 12343899, 46264257, 172305837, 638039136, 2350109736, 8613851832, 31428857611, 114187160631, 413222547846, 1489829356657, 5352683946903, 19167988920930, 68427472477338, 243559693397025

Offset: 0

Vaclav Kotesovec, Mar 16 2015

nonn

In general, if g.f. = Product_{j>=1} (1+x^j)^(k^j), then a(n) ~ k^n * exp(2*sqrt(n) - 1/2 - c(k)) / (2 * sqrt(Pi) * n^(3/4)), where c(k) = Sum_{m>=2} (-1)^m/(m*(k^(m-1)-1)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of sequence A034691
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

Cf. A102866, A144067, A144074.

Column k=3 of A292804.

nmax=30; CoefficientList[Series[Product[(1+x^k)^(3^k),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 3^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(3^(m-1)-1)) = 0.215985336303958581708278160877115129... .

A252654 Number of multisets of nonempty words with a total of n letters over n-ary alphabet.

Original entry on oeis.org

1, 1, 7, 64, 843, 13876, 276792, 6438797, 170938483, 5091463423, 167965714273, 6074571662270, 238837895468954, 10138497426332796, 461941179848628434, 22478593443737857695, 1163160397700757351363, 63760710281671647692688, 3690276585886363643056992

Offset: 0

Alois P. Heinz, Dec 19 2014

nonn

			a(2) = 7: {aa}, {ab}, {ba}, {bb}, {a,a}, {a,b}, {b,b}.

Alois P. Heinz, Table of n, a(n) for n = 0..350

Main diagonal of A144074.

Cf. A292805, A292873.

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
       d*k^d, d=divisors(j)) *A(n-j, k), j=1..n)/n)
    end:
a:= n-> A(n$2):
seq(a(n), n=0..25);

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*k^d, {d, Divisors[j]}]*A[n - j, k], {j, 1, n}]/n];
a[n_] := A[n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

a(n) = [x^n] Product_{j>=1} 1/(1-x^j)^(n^j).
a(n) = n-th term of the Euler transform of the powers of n.
a(n) ~ n^(n-3/4) * exp(2*sqrt(n) - 1/2) / (2*sqrt(Pi)). - Vaclav Kotesovec, Mar 14 2015
a(n) = [x^n] exp(n*Sum_{k>=1} x^k/(k*(1 - n*x^k))). - Ilya Gutkovskiy, Nov 20 2018

New name from comment by Alois P. Heinz, Sep 21 2018

A144067 Euler transform of powers of 3.

Original entry on oeis.org

1, 3, 15, 64, 276, 1137, 4648, 18585, 73494, 286834, 1108470, 4243128, 16111333, 60718488, 227302086, 845689753, 3128786415, 11515509603, 42179651417, 153808740042, 558532554942, 2020325112767, 7281212274165, 26151068072301, 93618849857345, 334119804933861

Offset: 0

Alois P. Heinz, Sep 09 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
N. J. A. Sloane, Transforms

3rd column of A144074. Row sums of A275414.

Cf. A256142.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^k)^(3^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:=n-> etr(j->3^j)(n): seq(a(n), n=0..40);

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; a[n_] := etr[Function[3^#]][n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)
CoefficientList[Series[Product[1/(1-x^k)^(3^k), {k, 1, 30}], {x, 0, 30}], x] (* G. C. Greubel, Nov 09 2018 *)

m=30; x='x+O('x^m); Vec(prod(k=1,m,1/(1-x^k)^(3^k))) \\ G. C. Greubel, Nov 09 2018

G.f.: Product_{j>0} 1/(1-x^j)^(3^j).
a(n) ~  3^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(3^(m-1)-1)) = 0.3047484092142751906436952201501007636114175... . - Vaclav Kotesovec, Mar 14 2015
G.f.: exp(3*Sum_{k>=1} x^k/(k*(1 - 3*x^k))). - Ilya Gutkovskiy, Nov 09 2018

A144068 Euler transform of powers of 4.

Original entry on oeis.org

1, 4, 26, 148, 843, 4632, 25124, 133784, 703553, 3655340, 18800886, 95819580, 484416675, 2431094352, 12120072472, 60058765072, 295959923287, 1450980481036, 7079894939166, 34393241899772, 166390593502701, 801877654792696, 3850469199935412, 18426281811165880

Offset: 0

Alois P. Heinz, Sep 09 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
N. J. A. Sloane, Transforms

4th column of A144074.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^k)^(4^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:=n-> etr(j->4^j)(n): seq(a(n), n=0..40);

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a[n_] := etr[Function[4^#]][n]; Table[ a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)
CoefficientList[Series[Product[1/(1-x^k)^(4^k), {k, 1, 30}], {x, 0, 30}], x] (* G. C. Greubel, Nov 09 2018 *)

m=30; x='x+O('x^m); Vec(prod(k=1,m,1/(1-x^k)^(4^k))) \\ G. C. Greubel, Nov 09 2018

G.f.: Product_{j>0} 1/(1-x^j)^(4^j).
a(n) ~  4^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(4^(m-1)-1)) = 0.1938490811676466793200632998157568919969827... . - Vaclav Kotesovec, Mar 14 2015
G.f.: exp(4*Sum_{k>=1} x^k/(k*(1 - 4*x^k))). - Ilya Gutkovskiy, Nov 09 2018

A144069 Euler transform of powers of 5.

Original entry on oeis.org

1, 5, 40, 285, 2020, 13876, 93885, 624480, 4100470, 26609290, 170940381, 1088260190, 6872684570, 43088845030, 268374618310, 1661512492031, 10229763359245, 62663268647185, 382039881168240, 2318974249801205, 14018511922088296, 84418983571948025

Offset: 0

Alois P. Heinz, Sep 09 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
N. J. A. Sloane, Transforms

5th column of A144074.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^k)^(5^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:=n-> etr(j->5^j)(n): seq(a(n), n=0..40);

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; a[n_] := etr[Function[5^#]][n]; Table[ a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)
CoefficientList[Series[Product[1/(1-x^k)^(5^k), {k, 1, 30}], {x, 0, 30}], x] (* G. C. Greubel, Nov 09 2018 *)

m=30; x='x+O('x^m); Vec(prod(k=1,m,1/(1-x^k)^(5^k))) \\ G. C. Greubel, Nov 09 2018

G.f.: Product_{j>0} 1/(1-x^j)^(5^j).
a(n) ~  5^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(5^(m-1)-1)) = 0.1412899716579209220312645657307029151422082... . - Vaclav Kotesovec, Mar 14 2015
G.f.: exp(5*Sum_{k>=1} x^k/(k*(1 - 5*x^k))). - Ilya Gutkovskiy, Nov 09 2018

A144070 Euler transform of powers of 6.

Original entry on oeis.org

1, 6, 57, 488, 4140, 34128, 276792, 2208312, 17389710, 135354340, 1042965042, 7964675400, 60337114778, 453795079932, 3390657365970, 25182770127240, 186007882964211, 1366948744701066, 9998341947058175, 72811720605519840, 528078809473488744, 3815340122599096360

Offset: 0

Alois P. Heinz, Sep 09 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

6th column of A144074.

Cf. A000400 (powers of 6).

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:=n-> etr(j->6^j)(n): seq(a(n), n=0..40);

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; a[n_] := etr[Function[6^#]][n]; Table[ a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

G.f.: Product_{j>0} 1/(1-x^j)^(6^j).
a(n) ~  6^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(6^(m-1)-1)) = 0.1108660629759785875628164141261367036457657... . - Vaclav Kotesovec, Mar 14 2015
G.f.: exp(6*Sum_{k>=1} x^k/(k*(1 - 6*x^k))). - Ilya Gutkovskiy, Nov 10 2018

A144071 Euler transform of powers of 7.

Original entry on oeis.org

1, 7, 77, 770, 7609, 73178, 691971, 6438797, 59131499, 536802112, 4824305213, 42970458839, 379692684987, 3330902681785, 29030038318212, 251498296181846, 2166886679835829, 18575273870841254, 158486917413708492, 1346334588169264925, 11390431451798171304

Offset: 0

Alois P. Heinz, Sep 09 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

7th column of A144074.

Cf. A000420 (powers of 7).

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:=n-> etr(j->7^j)(n): seq(a(n), n=0..40);

nmax = 20; CoefficientList[Series[Product[1/(1-x^j)^(7^j), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 14 2015 *)

G.f.: Product_{j>0} 1/(1-x^j)^(7^j).
a(n) ~  7^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(7^(m-1)-1)) = 0.0911034105381918017167778099460538483167631... . - Vaclav Kotesovec, Mar 14 2015
G.f.: exp(7*Sum_{k>=1} x^k/(k*(1 - 7*x^k))). - Ilya Gutkovskiy, Nov 10 2018

cp's OEIS Frontend

A034899 Euler transform of powers of 2 [ 2,4,8,16,... ].

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A257740 Number T(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet such that all k letters occur at least once in the multiset; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A292804 Number A(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A256142 G.f.: Product_{j>=1} (1+x^j)^(3^j).

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Mathematica

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A252654 Number of multisets of nonempty words with a total of n letters over n-ary alphabet.

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Maple

Mathematica

Formula

Extensions

A144067 Euler transform of powers of 3.

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Magma

Maple

Mathematica

PARI

Formula

A144068 Euler transform of powers of 4.

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