A275959 -id:A275959 - cp's OEIS frontend

A225901 Write n in factorial base, then replace each nonzero digit d of radix k with k-d.

0, 1, 4, 5, 2, 3, 18, 19, 22, 23, 20, 21, 12, 13, 16, 17, 14, 15, 6, 7, 10, 11, 8, 9, 96, 97, 100, 101, 98, 99, 114, 115, 118, 119, 116, 117, 108, 109, 112, 113, 110, 111, 102, 103, 106, 107, 104, 105, 72, 73, 76, 77, 74, 75, 90, 91, 94, 95, 92, 93, 84, 85, 88, 89, 86, 87, 78, 79, 82, 83, 80, 81, 48, 49, 52, 53, 50, 51, 66, 67, 70, 71, 68

Offset: 0

Paul Tek, May 20 2013

Analogous to A004488 or A048647 for the factorial base.
A self-inverse permutation of the natural numbers.
From Antti Karttunen, Aug 16-29 2016: (Start)
Consider the following way to view a factorial base representation of nonnegative integer n. For each nonzero digit d_i present in the factorial base representation of n (where i is the radix = 2.. = one more than 1-based position from the right), we place a pebble to the level (height) d_i at the corresponding column i of the triangular diagram like below, while for any zeros the corresponding columns are left empty:
.
Level
  6        o
          ─ ─
  5        . .
          ─ ─ ─
  4        . . .
          ─ ─ ─ ─
  3        . . . .
          ─ ─ ─ ─ ─
  2        . . o . .
          ─ ─ ─ ─ ─ ─
  1        . o . . o o
          ─ ─ ─ ─ ─ ─ ─
  Radix:   7 6 5 4 3 2
  Digits:  6 1 2 0 1 1 = A007623(4491)
Instead of levels, we can observe on which "slope" each pebble (nonzero digit) is located at. Formally, the slope of nonzero digit d_i with radix i is (i - d_i). Thus in above example, both the most significant digit (6) and the least significant 1 are on slope 1 (called "maximal slope", because it contains digits that are maximal allowed in those positions), while the second 1 from the right is on slope 2 ("submaximal slope").
This involution (A225901) sends each nonzero digit at level k to the slope k (and vice versa) by flipping such a diagram by the shallow diagonal axis that originates from the bottom right corner. Thus, from above diagram we obtain:
Slope (= digit's radix - digit's value)
   1
   2 .
   3 .  .╲
   4 .  .╲o╲
   5 .  .╲.╲.╲
   6 .  .╲.╲o╲.╲
     .  .╲.╲.╲.╲o╲
        o╲.╲.╲.╲.╲o╲
        -----------------
        1  5  3 0  2  1  = A007623(1397)
and indeed, a(4491) = 1397 and a(1397) = 4491.
Thus this permutation maps between polynomial encodings A275734 & A275735 and all the respective sequences obtained from them, where the former set of sequences are concerned with the "slopes" and the latter set with the "levels" of the factorial base representation. See the Crossrefs section.
Sequences A231716 and A275956 are closed with respect to this sequence, in other words, for all n, a(A231716(n)) is a term of A231716 and a(A275956(n)) is a term of A275956.
(End)

			a(1000) = a(1*6! + 2*5! + 1*4! + 2*3! + 2*2!) = (7-1)*6! + (6-2)*5! + (5-1)*4! + (4-2)*3! + (3-2)*2! = 4910.
a(1397) = a(1*6! + 5*5! + 3*4! + 0*3! + 2*2! + 1*1!) = (7-1)*6! + (6-5)*5! + (5-3)*4! + (3-2)*2! + (2-1)*1! = 4491.

Cf. A000142, A007623, A004488, A048647, A001563, A007489, A257684, A257687, A276091, A275736, A276149.
Cf. A275959 (fixed points), A231716, A275956.
Cf. A153880 & A255411.
Cf. also A275734 & A275735, A275952 & A275954.
This involution maps between the following sequences related to "levels" and "slopes" (see comments): A275806 <--> A060502, A257511 <--> A260736, A264990 <--> A275811, A275729 <--> A275728, A275948 <--> A275946, A275949 <--> A275947, A275964 <--> A275962, A059590 <--> A276091.
Related permutations: A275957, A275958, A275835, A275836, A275837, A275838, A276011, A276012.

b = MixedRadix[Reverse@ Range[2, 12]]; Table[FromDigits[Map[Boole[# > 0] &, #] (Reverse@ Range[2, Length@ # + 1] - #), b] &@ IntegerDigits[n, b], {n, 0, 82}] (* Version 10.2, or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Range[# + 1] <= n &]; Most@ Rest[a][[All, 1]] /. {} -> {0}]; g[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Range@ Range[0, Length@ w]], Reverse@ Append[w, 0]}]; Table[g[Map[Boole[# > 0] &, #] (Reverse@ Range[2, Length@ # + 1] - #)] &@ f@ n, {n, 0, 82}] (* Michael De Vlieger, Aug 29 2016 *)

a(n)=my(s=0,d,k=2);while(n,d=n%k;n=n\k;if(d,s=s+(k-d)*(k-1)!);k=k+1);return(s)

from sympy import factorial as f
def a(n):
    s=0
    k=2
    while(n):
        d=n%k
        n=(n//k)
        if d: s=s+(k - d)*f(k - 1)
        k+=1
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 19 2017

(define (A225901 n) (let loop ((n n) (z 0) (m 2) (f 1)) (cond ((zero? n) z) (else (loop (quotient n m) (if (zero? (modulo n m)) z (+ z (* f (- m (modulo n m))))) (+ 1 m) (* f m))))))
;; One implementing the first recurrence, with memoization-macro definec:
(definec (A225901 n) (if (zero? n) n (+ (A276091 (A275736 n)) (A153880 (A225901 (A257684 n))))))
;; Antti Karttunen, Aug 29 2016

From Antti Karttunen, Aug 29 2016: (Start)
a(0) = 0; for n >= 1, a(n) = A276091(A275736(n)) + A153880(a(A257684(n))).
or, for n >= 1, a(n) = A276149(n) + a(A257687(n)).
(End)
Other identities. For n >= 0:
a(n!) = A001563(n).
a(n!-1) = A007489(n-1).
From Antti Karttunen, Aug 16 2016: (Start)
A275734(a(n)) = A275735(n) and vice versa, A275735(a(n)) = A275734(n).
A060130(a(n)) = A060130(n). [The flip preserves the number of nonzero digits.]
A153880(n) = a(A255411(a(n))) and A255411(n) = a(A153880(a(n))). [This involution conjugates between the two fundamental factorial base shifts.]
a(n) = A257684(a(A153880(n))) = A266193(a(A255411(n))). [Follows from above.]
A276011(n) = A273662(a(A273670(n))).
A276012(n) = A273663(a(A256450(n))).
(End)

A059590 Numbers obtained by reinterpreting base-2 representation of n in the factorial base: a(n) = Sum_{k>=0} A030308(n,k)*A000142(k+1).

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 24, 25, 26, 27, 30, 31, 32, 33, 120, 121, 122, 123, 126, 127, 128, 129, 144, 145, 146, 147, 150, 151, 152, 153, 720, 721, 722, 723, 726, 727, 728, 729, 744, 745, 746, 747, 750, 751, 752, 753, 840, 841, 842, 843, 846, 847, 848, 849, 864, 865

Offset: 0

Henry Bottomley, Jan 24 2001

Numbers that are sums of distinct factorials (0! and 1! not treated as distinct).
Complement of A115945; A115944(a(n)) > 0; A115647 is a subsequence. - Reinhard Zumkeller, Feb 02 2006
A115944(a(n)) = 1. - Reinhard Zumkeller, Dec 04 2011
From Tilman Piesk, Jun 04 2012: (Start)
The inversion vector (compare A007623) of finite permutation a(n) (compare A055089, A195663) has only zeros and ones. Interpreted as a binary number it is 2*n (or n when the inversion vector is defined without the leading 0).
The inversion set of finite permutation a(n) interpreted as a binary number (compare A211362) is A211364(n).
(End)

			128 is in the sequence since 5! + 3! + 2! = 128.
a(22) = 128. a(22) = a(6) + (1 + floor(log(16) / log(2)))! = 8 + 5! = 128. Also, 22 = 10110_2. Therefore, a(22) = 1 * 5! + 0 * 4! + 1 * 3! + 1 + 2! + 0 * 0! = 128. - _David A. Corneth_, Aug 21 2016

Reinhard Zumkeller (terms 0..500) & Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to factorial base representation
Index entries for sequences related to factorial numbers

Cf. A007088, A007623, A014597, A051760, A051761, A059589.
Indices of zeros in A257684.
Cf. A275736 (left inverse).
Cf. A025494, A060112 (subsequences).
Cf. A153880, A225901.
Subsequence of A060132, A256450 and A275804.
Other sequences that are built by replacing 2^k in the binary representation with other numbers: A029931 (naturals), A089625 (primes), A022290 (Fibonacci), A197433 (Catalans), A276091 (n*n!), A275959 ((2n)!/2). Cf. also A276082 & A276083.

import Data.List (elemIndices)
a059590 n = a059590_list !! n
a059590_list = elemIndices 1 $ map a115944 [0..]
-- Reinhard Zumkeller, Dec 04 2011

[seq(bin2facbase(j),j=0..64)]; bin2facbase := proc(n) local i; add((floor(n/(2^i)) mod 2)*((i+1)!),i=0..floor_log_2(n)); end;
floor_log_2 := proc(n) local nn,i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end;
# next Maple program:
a:= n-> (l-> add(l[j]*j!, j=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..57);  # Alois P. Heinz, Aug 12 2025

a[n_] :=  Reverse[id = IntegerDigits[n, 2]].Range[Length[id]]!; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jun 19 2012, after Philippe Deléham *)

a(n) = if(n>0, a(n-msb(n)) + (1+logint(n,2))!, 0)
msb(n) = 2^#binary(n)>>1
{my(b = binary(n)); sum(i=1,#b,b[i]*(#b+1-i)!)} \\ David A. Corneth, Aug 21 2016

def facbase(k, f):
    return sum(f[i] for i, bi in enumerate(bin(k)[2:][::-1]) if bi == "1")
def auptoN(N): # terms up to N factorial-base digits; 13 generates b-file
    f = [factorial(i) for i in range(1, N+1)]
    return list(facbase(k, f) for k in range(2**N))
print(auptoN(5)) # Michael S. Branicky, Oct 15 2022

G.f. 1/(1-x) * Sum_{k>=0} (k+1)!*x^2^k/(1+x^2^k). - Ralf Stephan, Jun 24 2003
a(n) = Sum_{k>=0} A030308(n,k)*A000142(k+1). - Philippe Deléham, Oct 15 2011
From Antti Karttunen, Aug 19 2016: (Start)
a(0) = 0, a(2n) = A153880(a(n)), a(2n+1) = 1+A153880(a(n)).
a(n) = A225901(A276091(n)).
a(n) = A276075(A019565(n)).
a(A275727(n)) = A276008(n).
A275736(a(n)) = n.
A276076(a(n)) = A019565(n).
A007623(a(n)) = A007088(n).
(End)
a(n) = a(n - mbs(n)) + (1 + floor(log(n) / log(2)))!. - David A. Corneth, Aug 21 2016

Name changed (to emphasize the functional nature of the sequence) with the old definition moved to the comments by Antti Karttunen, Aug 21 2016

A276091 Numbers obtained by reinterpreting base-2 representation of n in A001563-base (A276326): a(n) = Sum_{k>=0} A030308(n,k)*A001563(k+1).

Original entry on oeis.org

0, 1, 4, 5, 18, 19, 22, 23, 96, 97, 100, 101, 114, 115, 118, 119, 600, 601, 604, 605, 618, 619, 622, 623, 696, 697, 700, 701, 714, 715, 718, 719, 4320, 4321, 4324, 4325, 4338, 4339, 4342, 4343, 4416, 4417, 4420, 4421, 4434, 4435, 4438, 4439, 4920, 4921, 4924, 4925, 4938, 4939, 4942, 4943, 5016, 5017, 5020, 5021, 5034, 5035, 5038, 5039, 35280, 35281

Offset: 0

Antti Karttunen, Aug 19 2016

Numbers that are sums of distinct terms of A001563.
A number is included if and only if all the nonzero digits in its factorial base representation (A007623) are maximal allowed in those digit positions, thus this sequence gives all numbers n for which A060130(n) = A260736(n).
Numbers n for which A276328(n) = A276337(n), thus from 1 onward the positions of ones in A276336.
Conjectured also to give all numbers n for which A255411(n) = A276340(n) (thus zeros of A276339).

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to factorial base representation

Cf. A000120, A000142, A001563, A030308, A059590, A060130, A260736, A225901, A255411, A275959, A276082, A276083, A276090, A276326, A276328, A276336, A276337, A276339, A276340.

Table[Total[Times @@@ Transpose@ {Map[# #! &, Range@ Length@ #], Reverse@ #}] &@ IntegerDigits[n, 2], {n, 64}] (* Michael De Vlieger, Aug 31 2016 *)

from sympy import factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return 0 if n==0 else sum(int(y[i])*f(i + 1) for i in range(len(y)))
def a(n): return 0 if n==0 else a255411(a(n//2)) if n%2==0 else 1 + a255411(a((n - 1)//2))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017

;; This is a standalone program:
(define (A276091 n) (let loop ((n n) (s 0) (f 1) (i 2)) (cond ((zero? n) s) ((even? n) (loop (/ n 2) s (* i f) (+ 1 i))) (else (loop (/ (- n 1) 2) (+ s (* (- i 1) f)) (* i f) (+ 1 i))))))
;; This implements one of the given recurrences:
(definec (A276091 n) (cond ((zero? n) n) ((even? n) (A255411 (A276091 (/ n 2)))) (else (+ 1 (A255411 (A276091 (/ (- n 1) 2)))))))
;; Alternatively, we can use A276340 in place of A255411:
(definec (A276091 n) (cond ((zero? n) n) ((even? n) (A276340 (A276091 (/ n 2)))) (else (+ 1 (A276340 (A276091 (/ (- n 1) 2)))))))

a(0) = 0, a(2n) = A255411(a(n)), a(2n+1) = 1+A255411(a(n)).
a(0) = 0, a(2n) = A276340(a(n)), a(2n+1) = 1+A276340(a(n)).
Other identities. For all n >= 0:
a(n) = A225901(A059590(n)).
a(n) = A276090(A275959(n)).
A276328(a(n)) = A276337(a(n)) = A000120(n).

Name changed (to emphasize the functional nature of the sequence) with the original definition moved to the comments by Antti Karttunen, Sep 01 2016

A276082 a(0) = 0, a(2n) = A153880(a(n)), a(2n+1) = 1+A255411(a(n)).

Original entry on oeis.org

0, 1, 2, 5, 6, 13, 14, 23, 24, 49, 50, 77, 54, 85, 86, 119, 120, 241, 242, 365, 246, 373, 374, 503, 264, 409, 410, 557, 414, 565, 566, 719, 720, 1441, 1442, 2165, 1446, 2173, 2174, 2903, 1464, 2209, 2210, 2957, 2214, 2965, 2966, 3719, 1560, 2401, 2402, 3245, 2406, 3253, 3254, 4103, 2424, 3289, 3290, 4157, 3294, 4165, 4166, 5039, 5040, 10081

Offset: 0

Antti Karttunen, Aug 21 2016

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to factorial base representation

Cf. A153880, A255411

Cf. also A059590, A275959, A276091, A225901, A276083.

from sympy import factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return 0 if n==0 else sum([int(y[i])*f(i + 1) for i in range(len(y))])
def a153880(n):
    x=(str(a007623(n)) + '0')[::-1]
    return 0 if n==0 else sum([int(x[i])*f(i + 1) for i in range(len(x))])
def a(n): return 0 if n==0 else a153880(a(n//2)) if n%2==0 else 1 + a255411(a((n - 1)//2))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017

a(0) = 0, a(2n) = A153880(a(n)), a(2n+1) = 1+A255411(a(n)).
Other identities. For all n >= 0:
a(n) = A225901(A276083(n)).

A276083 a(0) = 0, a(2n) = A255411(a(n)), a(2n+1) = 1+A153880(a(n)).

Original entry on oeis.org

0, 1, 4, 3, 18, 13, 16, 9, 96, 73, 76, 51, 90, 61, 64, 33, 600, 481, 484, 363, 498, 373, 376, 249, 576, 433, 436, 291, 450, 301, 304, 153, 4320, 3601, 3604, 2883, 3618, 2893, 2896, 2169, 3696, 2953, 2956, 2211, 2970, 2221, 2224, 1473, 4200, 3361, 3364, 2523, 3378, 2533, 2536, 1689, 3456, 2593, 2596, 1731, 2610, 1741, 1744, 873, 35280, 30241

Offset: 0

Antti Karttunen, Aug 21 2016

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to factorial base representation

Cf. A153880, A255411.

Cf. also A059590, A275959, A276091, A225901, A276082.

from sympy import factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return 0 if n==0 else sum([int(y[i])*f(i + 1) for i in range(len(y))])
def a153880(n):
    x=(str(a007623(n)) + '0')[::-1]
    return 0 if n==0 else sum([int(x[i])*f(i + 1) for i in range(len(x))])
def a(n): return 0 if n==0 else a255411(a(n//2)) if n%2==0 else 1 + a153880(a((n - 1)//2))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017

a(0) = 0, a(2n) = A255411(a(n)), a(2n+1) = 1+A153880(a(n)).
Other identities. For all n >= 0:
a(n) = A225901(A276082(n)).

A275956 Numbers n for which A060502(n) = A275806(n); numbers whose factorial base representation has an equal number of distinct nonzero digits and occupied slopes.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 11, 12, 13, 15, 17, 18, 20, 21, 24, 28, 29, 36, 38, 42, 43, 48, 49, 50, 53, 55, 56, 58, 59, 62, 66, 68, 69, 70, 72, 73, 75, 76, 78, 80, 82, 83, 88, 91, 92, 93, 94, 96, 98, 99, 102, 103, 108, 112, 120, 124, 125, 132, 134, 138, 139, 166, 167, 168, 174, 186, 187, 190, 191, 192, 194, 196, 197, 205, 207, 208, 209, 214, 215, 216, 217, 226

Offset: 0

Antti Karttunen, Aug 16 2016

Nonnegative integer n is included here iff the number of distinct nonzero digits elements present in its factorial base representation is equal to the number of distinct elements in a multiset [(i_x - d_x) | where d_x ranges over each (not all necessarily distinct) nonzero digit present and i_x is that digit's position from the right].

If n is included, then A153880(n), A255411(n) and A225901(n) are also included.

			11 ("121" in factorial base) is included, as there are two occupied slopes (3-1 = 2 and 2-2 = 1-1 = 0) and also two distinct nonzero digits, namely 1 and 2.
59 ("2121" in factorial base) is included, as there are two occupied slopes (4-2 = 3-1 and 2-2 = 1-1) and also two distinct nonzero digits, namely 1 and 2.
226 ("14120") is included, as there are three occupied slopes (5-1 = 4, 4-4 = 2-2 = 0, 3-1 = 2) and also three distinct nonzero digits, 1, 2 and 4.

Antti Karttunen, Table of n, a(n) for n = 0..12558
Index entries for sequences related to factorial base representation

Cf. A060502, A275806.
Cf. A153880, A255411, A225901.
Cf. A000142, A275959 (subsequences).

A276089 Factorial base representation of n is aerated (and then converted back to decimal): for n = sum_{i=1..} digit(i)i! (with each digit(i) <= i), a(n) = sum_{i=1..} digit(i)(2i-1)!.

Original entry on oeis.org

0, 1, 6, 7, 12, 13, 120, 121, 126, 127, 132, 133, 240, 241, 246, 247, 252, 253, 360, 361, 366, 367, 372, 373, 5040, 5041, 5046, 5047, 5052, 5053, 5160, 5161, 5166, 5167, 5172, 5173, 5280, 5281, 5286, 5287, 5292, 5293, 5400, 5401, 5406, 5407, 5412, 5413, 10080, 10081, 10086, 10087, 10092, 10093, 10200, 10201, 10206, 10207, 10212, 10213, 10320, 10321

Offset: 0

Antti Karttunen, Aug 19 2016

Here "aeration" means inserting zeros between the digits of factorial base representation of n. See the examples.

			   n A007623(n)  "aerated"   and converted back to decimal gives
                             a(n)
   0       0         0         0
   1       1         1         1
   2      10       100         6
   3      11       101         7
   4      20       200        12
   5      21       201        13
   6     100     10000       120
   7     101     10001       121
   8     110     10100       126
   9     111     10101       127
  10     120     10200       132
  11     121     10201       133
  12     200     20000       240
  13     201     20001       241
  14     210     20100       246
  15     211     20101       247
  16     220     20200       252
  17     221     20201       253
  18     300     30000       360
  19     301     30001       361
  20     310     30100       366
  21     311     30101       367
  22     320     30200       372
  23     321     30201       373

Antti Karttunen, Table of n, a(n) for n = 0..5039
Index entries for sequences related to factorial base representation

Cf. A000142, A007623, A084558
Cf. A276090 (a left inverse).
Cf. A275959 (subsequence).

;; Standalone program:
(define (A276089 n) (let loop ((n n) (s 0) (f 1) (i 2) (j 2)) (if (zero? n) s (let ((d (modulo n i))) (loop (/ (- n d) i) (+ s (* f d)) (* j (+ 1 j) f) (+ 1 i) (+ 2 j))))))

When n = sum_{i=1..A084558(n)} d(i)*i! (which is unique representation as long as each d(i) <= i) then a(n) = sum_{i=1..A084558(n)} d(i)*(2i-1)!, where A084558(n) gives the length of factorial base representation of n.
Other identities. For all n >= 0:
A276090(a(n)) = n.

cp's OEIS Frontend

A225901 Write n in factorial base, then replace each nonzero digit d of radix k with k-d.

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A059590 Numbers obtained by reinterpreting base-2 representation of n in the factorial base: a(n) = Sum_{k>=0} A030308(n,k)*A000142(k+1).

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A276091 Numbers obtained by reinterpreting base-2 representation of n in A001563-base (A276326): a(n) = Sum_{k>=0} A030308(n,k)*A001563(k+1).

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A276082 a(0) = 0, a(2n) = A153880(a(n)), a(2n+1) = 1+A255411(a(n)).

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A276083 a(0) = 0, a(2n) = A255411(a(n)), a(2n+1) = 1+A153880(a(n)).

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A275956 Numbers n for which A060502(n) = A275806(n); numbers whose factorial base representation has an equal number of distinct nonzero digits and occupied slopes.

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A276089 Factorial base representation of n is aerated (and then converted back to decimal): for n = sum_{i=1..} digit(i)*i! (with each digit(i) <= i), a(n) = sum_{i=1..} digit(i)*(2i-1)!.

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A276089 Factorial base representation of n is aerated (and then converted back to decimal): for n = sum_{i=1..} digit(i)i! (with each digit(i) <= i), a(n) = sum_{i=1..} digit(i)(2i-1)!.