A005130 -id:A005130 - cp's OEIS frontend

A049503 a(n) = A005130(n)^2.

1, 1, 4, 49, 1764, 184041, 55294096, 47675849104, 117727187246656, 831443906113411600, 16779127803917965290000, 966945347924006310543140625, 159045186822042363450404006250000, 74638947576233124529271587010756250000, 99910846988474589225795290311922220324000000

Offset: 0

N. J. A. Sloane

Expansion of generating function A_{QT}^(1)(4n).

a(n) is the number of cyclically symmetric and self-complementary plane partitions in a (2n)-cube. - Peter J. Taylor, Jun 17 2015

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.16), p. 199.

Seiichi Manyama, Table of n, a(n) for n = 0..66
M. Ciucu, The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions, J. Combin. Theory Ser. A 86 (1999), 382-389.
G. Kuperberg, Symmetries of plane partitions and the permanent-determinant method, J. Comb. Theory Ser. A, 68 (1994), 115-151. [From _Peter J. Taylor_, Jun 17 2015]
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001. [Th. 5].
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.
Wikipedia, Plane partition

Cf. A005130.

[n eq 0 select 1 else &*[(Factorial(3*k+1)/Factorial(n+k))^2: k in [0..n-1]]: n in [0..15]]; // Bruno Berselli, Jun 23 2015

f[n_]:=Product[((3 k + 1)!/(n + k)!)^2, {k, 0, n-1}]; Table[f[n], {n, 0, 15}] (* Vincenzo Librandi, Jun 18 2015 *)

a(n) = 2^n*matdet(matrix(n, n, i, j, i--; j--; binomial(i+j, 2*i-j)/2+binomial(i+j, 2*i-j-1))); \\ Michel Marcus, Jun 18 2015

from math import prod, factorial
def A049503(n): return (prod(factorial(3*k+1) for k in range(n))//prod(factorial(n+k) for k in range(n)))**2 # Chai Wah Wu, Feb 02 2022

a(n) = 2^n * det U(n), where U(n) is the n X n matrix with entry (i, j) equal to binomial(i+j, 2*i-j)/2 + binomial(i+j, 2*i-j-1). [Ciucu]

A160708 Convolution triangle by rows, row sums = the Robbins sequence, A005130 starting with offset 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 18, 3, 3, 18, 192, 18, 9, 18, 192, 3472, 192, 54, 54, 192, 3472, 104964, 3472, 576, 324, 576, 3472, 104964, 5606272, 104964, 10416, 3456, 3456, 10416, 104964, 5306272

Offset: 1

Gary W. Adamson, May 24 2009

The terms are not integral in general, see A160707. - Joerg Arndt, Jan 02 2019
Row sums = the Robbins sequence A005130, starting with offset 1: (1, 2, 7, 42, 429,...).
Right and left borders = A160707, the convolution square root of A005130.

			First few rows of the triangle =
1;
1, 1;
3, 1, 3;
18, 3, 3, 18;
192, 18, 9, 18, 192;
3472, 192, 54, 54, 192, 3472;
104964, 3472, 576, 324, 576, 3472, 104964;
5606272, 104964, 10416, 3456, 3456, 10416, 104964, 5306272;
...
Example: row 5 = (192, 18, 9, 18, 192) = (192, 18, 3, 1, 1) * (1, 1, 3, 18, 192); where A005130(5) = 429 = (192 + 18 + 9 + 18 + 192).

Cf. A005130, A160707

Let M = an infinite lower triangular Toeplitz matrix with A160707 in every column: (1, 1, 3, 18, 192, 5472,...); where A160707 = the convolution square root of the Robbins sequence: (1, 2, 7, 42, 429, 7436,...). Let Q = an infinite lower triangular matrix with (1, 1, 3, 18, 192,...) as the main diagonal and the rest zeros. Triangle A160708 = M * Q.

A194827 2-adic valuation of the number of n X n Alternating Sign Matrices (A005130(n)).

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 2, 3, 2, 2, 0, 2, 2, 4, 4, 5, 4, 4, 2, 2, 0, 3, 4, 6, 6, 7, 6, 8, 8, 10, 10, 11, 10, 10, 8, 8, 6, 7, 6, 6, 4, 3, 0, 3, 4, 7, 8, 10, 10, 11, 10, 11, 10, 13, 14, 16, 16, 17, 16, 18, 18, 20, 20, 21, 20, 20, 18, 18, 16, 17, 16, 16, 14, 13, 10, 11, 10, 11, 10, 10

Offset: 1

R. J. Mathar, Sep 03 2011

Kenny Lau, Table of n, a(n) for n = 1..9999
Clemens Heuberger and Helmut Prodinger, A precise description of the p-adic valuation of the number of alternating sign matrices, Intl. J. Numb. Th., Vol. 7, No. 1 (2011), pp. 57-69.

Cf. A000120, A005130, A007814, A227833.

Sp := proc(n,p) add(d,d=convert(n,base,p)) ; end proc:
nuA005130 := proc(n,p) add(Sp(n+j,p),j=0..n-1)-add(Sp(3*j+1,p),j=0..n-1) ; %/(p-1) ; end proc:
A194827 := proc(n) nuA005130(n,2) ; end proc:

s[n_] := DigitCount[n, 2, 1]; a[0] = 0; a[n_] := a[n] = a[n - 1] + s[2*n - 2] + s[2*n - 1] - s[n - 1] - s[3*n - 2]; Array[a, 100] (* Amiram Eldar, Feb 21 2021 *)

# a(n) = prod(k=0, n-1, (3k+1)!/(n+k)!)
# a(n+1) = prod(k=0, n, (3k+1)!/(n+k+1)!)
# a(n+1) = prod(k=0, n, (3k+1)!/(n+k)!) prod(k=0, n, 1/(n+k+1))
# a(n+1)/a(n) = [(3n+1)!/(2n)!] [n!/(2n+1)!]
n=10000; N=3*n+1; val=[0]*(N+1); exp=2
while exp <= N:
    for j in range(exp,N+1,exp): val[j] += 1
    exp *= 2
fac_val=[0]*(N+1)
for i in range(N): fac_val[i+1] = fac_val[i] + val[i+1]
res=0
for i in range(1,n): print(i,res); res += fac_val[3*i+1] + fac_val[i] - fac_val[2*i] - fac_val[2*i+1]
# Kenny Lau, Jun 09 2018

a(n) = A007814(A005130(n)).

a(n) = a(n-1) + s(2*n-2) + s(2*n-1) - s(n-1) - s(3*n-2), where s(n) = A000120(n). - Amiram Eldar, Feb 21 2021

A227833 3-adic valuation of A005130(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 3, 5, 5, 3, 2, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 3, 5, 6, 6, 7, 8, 9, 11, 13, 15, 18, 18, 15, 13, 11, 9, 8, 7, 6, 6, 5, 3, 2, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 3, 5, 5, 3, 2, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 3, 5, 6, 6, 7, 8, 9, 11, 13, 15, 18, 19, 18, 18, 18, 18, 19

Offset: 0

N. J. A. Sloane, Aug 04 2013, based on a suggestion from Victor H. Moll

nonn

3^a(n) is the highest power of 3 dividing A005130(n).

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A005130, A007949, A053735, A194827.

A005130:=func; [Valuation(A005130(n),3): n in [0..100]]; // Bruno Berselli, Aug 05 2013

Mathematica

s[n_] := Plus @@ IntegerDigits[n, 3]; a[0] = 0; a[n_] := a[n] = a[n - 1] + (s[2*n - 2] + s[2*n - 1] - s[n - 1] - s[3*n - 2])/2; Array[a, 101, 0] (* Amiram Eldar, Feb 21 2021 *)

Formula

From Amiram Eldar, Feb 21 2021: (Start)

a(n) = A007949(A005130(n)).

a(n) = a(n-1) + (s(2*n-2) + s(2*n-1) - s(n-1) - s(3*n-2))/2, where s(n) = A053735(n). (End)

A160707 Convolution square root of the Robbins sequence, A005130, starting with offset 1.

Original entry on oeis.org

1, 1, 3, 18, 192, 3472, 104964, 5306272, 450215638, 64298445920

Offset: 1

Gary W. Adamson, May 24 2009

Terms are not integral beyond what's shown! - Joerg Arndt, Jan 02 2019

			Self-convolution = the Robbins sequence, A005130: (1, 2, 7, 42, 429, 7436, ...).
Example: A005130(4) = 42 = (1, 1, 3, 18) dot (18, 3, 1, 1) = (18 + 3 + 3 + 18).
The reversed procedure given (1, 2, 7, 42, ...) for a(4) = 18 = (1/2) * (42 - 2*a(3)) = (1/2) * 36 = 18.

Cf. A005130, A160708.

A173312 Partial sums of A005130.

Original entry on oeis.org

1, 2, 4, 11, 53, 482, 7918, 226266, 11076482, 922911942, 130457184642, 31226202037017, 12642538061714517, 8652026056359367017, 10004193381504526849017, 19539080428042781631746217

Offset: 0

Jonathan Vos Post, Feb 16 2010

nonn

Partial sums of Robbins numbers. Partial sums of the number of descending plane partitions whose parts do not exceed n. Partial sums of the number of n X n alternating sign matrices (ASM's). After 2, 11, 53, when is this partial sum again prime, as it is not again prime through a(32)?

			a(17) = 1 + 1 + 2 + 7 + 42 + 429 + 7436 + 218348 + 10850216 + 911835460 + 129534272700 + 31095744852375 + 12611311859677500 + 8639383518297652500 + 9995541355448167482000 + 19529076234661277104897200 + 64427185703425689356896743840 + 358869201916137601447486156417296.

Cf. A005130, A006366, A048601, also A003827, A005156, A005158, A005160-A005164, A050204, A049503, A160707, A160708.

Table[Sum[Product[(3 k + 1)!/(j + k)!, {k, 0, j - 1}], {j, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 26 2017 *)
Accumulate[Table[Product[(3k+1)!/(n+k)!,{k,0,n-1}],{n,0,20}]] (* Harvey P. Dale, Feb 06 2019 *)

a(n) = Sum_{i=0..n} A005130(i) = Sum_{i=0..n} Product_{k=0..i-1} (3k+1)!/(i+k)!. [corrected by Vaclav Kotesovec, Oct 26 2017]

a(n) ~ Pi^(1/3) * exp(1/36) * 3^(3*n^2/2 - 7/36) / (A^(1/3) * Gamma(1/3)^(2/3) * n^(5/36) * 2^(2*n^2 - 5/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 26 2017

A001462 Golomb's sequence: a(n) is the number of times n occurs, starting with a(1) = 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19

Offset: 1

N. J. A. Sloane

It is understood that a(n) is taken to be the smallest number >= a(n-1) which is compatible with the description.
In other words, this is the lexicographically earliest nondecreasing sequence of positive numbers which is equal to its RUNS transform. - N. J. A. Sloane, Nov 07 2018
Also called Silverman's sequence.
Vardi gives several identities satisfied by A001463 and this sequence.
We can interpret this sequence as a triangle: start with 1; 2,2; 3,3; and proceed by letting the row sum of row m-1 be the number of elements of row m. The partial sums of the row sums give 1, 5, 11, 38, 272, ... Conjecture: this proceeds as Lionel Levine's sequence A014644. See also A113676. - Floor van Lamoen, Nov 06 2005
A Golomb-type sequence, that is, one with the property of being a sequence of run length of itself, can be built over any sequence with distinct terms by repeating each term a corresponding number of times, in the same manner as a(n) is built over natural numbers. See cross-references for more examples. - Ivan Neretin, Mar 29 2015
From Amiram Eldar, Jun 19 2021: (Start)
Named after the American mathematician Solomon Wolf Golomb (1932-2016).
Guy (2004) called it "Golomb's self-histogramming sequence", while in previous editions of his book (1981 and 1994) he called it "Silverman's sequence" after David Silverman. (End)
a(n) is also the number of numbers that occur n times. - Leo Crabbe, Feb 15 2025

			a(1) = 1, so 1 only appears once. The next term is therefore 2, which means 2 appears twice and so a(3) is also 2 but a(4) must be 3. And so on.
G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 4*x^8 + ... - _Michael Somos_, Nov 07 2018

Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 10.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 66.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section E25, p. 347-348.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Altug Alkan, On a Generalization of Hofstadter's Q-Sequence: A Family of Chaotic Generational Structures, Complexity (2018), Article ID 8517125.
Joaquim Bruna, The golden ratio from a calculus point of view, Universitat Autònoma de Barcelona, Materials matemàtics (2023) Vol. 2023, No. 4.
Benoit Cloitre, N. J. A. Sloane and Matthew J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seq., Vol. 6 (2003), Article 03.2.2.
Benoit Cloitre, N. J. A. Sloane and Matthew J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Martin Gardner, Letter to N. J. A. Sloane, Jun 20 1991.
Solomon W. Golomb, Problem 5407, Amer. Math. Monthly, Vol. 73, No. 6 (1966), p. 674.
Jarosław Grytczuk, Another variation on Conway's recursive sequence, Discr. Math., Vol. 282, No. 1-3 (2004), pp. 149-161.
Brady Haran and Tony Padilla, Six Sequences, Numberphile video (2013).
Brady Haran and N. J. A. Sloane, Planing Sequences (Le Rabot), Numberphile video, June 2021.
Daniel Marcus and N. J. Fine, Solutions to Problem 5407, Amer. Math. Monthly, Vol. 74, No. 6 (1967), pp. 740-743.
Christian Perfect, Integer sequence reviews on Numberphile (or vice versa), 2013.
Y.-F. S. Petermann, On Golomb's self-describing sequence, J. Number Theory, Vol. 53, No. 1 (1995), pp. 13-24.
Y.-F. S. Petermann, On Golomb's self-describing sequence, II, Arch. Math. (Basel) 67 (1996), 473-477.
Y.-F. S. Petermann, Is the error term wild enough?, Analysis (Munich), Vol. 18 (1998), pp. 245-256.
Y.-F. S. Pétermann and Jean-Luc Rémy, Golomb's self-described sequence and functional-differential equations, Illinois J. Math., Vol. 42, No. 3 (1998), pp. 420-440.
Y.-F. S. Pétermann, Jean-Luc Rémy and Ilan Vardi, Discrete derivatives of sequences, Adv. in Appl. Math., Vol. 27 (2001), pp. 562-84.
Jean-Luc Rémy, Sur la suite autodécrite de Golomb, J. Number Theory, Vo. 66, No. 1 (1997), pp. 1-28.
Jim Sauerberg and Linghsueh Shu, The long and the short on counting sequences, Amer. Math. Monthly, Vol. 104, No. 4 (1997), pp. 306-317.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Seven Staggering Sequences.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970. (note that A1148 has now become A005282)
N. J. A. Sloane, Transforms.
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
Ilan Vardi, The error term in Golomb's sequence, J. Number Theory, Vol. 40 (1992), pp. 1-11. (See also the Math. Review, 93d:11103)
Eric Weisstein's World of Mathematics, Silverman's Sequence.
Index entries for "core" sequences
Index entries for sequences of the a(a(n)) = 2n family

Cf. A001463 (partial sums) and A262986 (start of first run of length n).
First differences are A088517.
Golomb-type sequences over various substrates (from Glen Whitney, Oct 12 2015):
A000002 and references therein (over periodic sequences),
A109167 (over nonnegative integers),
A080605 (over odd numbers),
A080606 (over even numbers),
A080607 (over multiples of 3),
A169682 (over primes),
A013189 (over squares),
A013322 (over triangular numbers),
A250983 (over integral sums of itself).
Applying "ee Rabot" to this sequence gives A319434.
See also A095114.

a001462 n = a001462_list !! (n-1)
a001462_list = 1 : 2 : 2 : g 3  where
   g x = (replicate (a001462 x) x) ++ g (x + 1)
-- Reinhard Zumkeller, Feb 09 2012

[ n eq 1 select 1 else 1+Self(n-Self(Self(n-1))) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

N:= 10000: A001462[1]:= 1: B[1]:= 1: A001462[2]:= 2:
for n from 2 while B[n-1] <= N do
  B[n]:= B[n-1] + A001462[n];
  for j from B[n-1]+1 to B[n] do A001462[j]:= n end do
end do:
seq(A001462[j],j=1..N); # Robert Israel, Oct 30 2012

a[1] = 1; a[n_] := a[n] = 1 + a[n - a[a[n - 1]]]; Table[ a[n], {n, 84}] (* Robert G. Wilson v, Aug 26 2005 *)
GolSeq[n_]:=Nest[(k = 0; Flatten[# /. m_Integer :> (ConstantArray[++k,m])]) &, {1, 2}, n]
GolList=Nest[(k = 0;Flatten[# /.m_Integer :> (ConstantArray[++k,m])]) &, {1, 2},7]; AGolList=Accumulate[GolList]; Golomb[n_]:=Which[ n <= Length[GolList], GolList[[n]], n <= Total[GolList],First[FirstPosition[AGolList, ?(# > n &)]], True, $Failed] (* _JungHwan Min, Nov 29 2015 *)

a = [1, 2, 2]; for(n=3, 20, for(i=1, a[n], a = concat(a, n))); a /* Michael Somos, Jul 16 1999 */

{a(n) = my(A, t, i); if( n<3, max(0, n), A = vector(n); t = A[i=2] = 2; for(k=3, n, A[k] = A[k-1] + if( t--==0, t = A[i++]; 1)); A[n])}; /* Michael Somos, Oct 21 2006 */

a=[0, 1, 2, 2]
for n in range(3, 21):a+=[n for i in range(1, a[n] + 1)]
a[1:] # Indranil Ghosh, Jul 05 2017

a(n) = phi^(2-phi)*n^(phi-1) + E(n), where phi is the golden number (1+sqrt(5))/2 (Marcus and Fine) and E(n) is an error term which Vardi shows is O( n^(phi-1) / log n ).
a(1) = 1; a(n+1) = 1 + a(n+1-a(a(n))). - Colin Mallows
a(1)=1, a(2)=2 and for a(1) + a(2) + ... + a(n-1) < k <= a(1) + a(2) + ... + a(n) we have a(k)=n. - Benoit Cloitre, Oct 07 2003
G.f.: Sum_{n>0} a(n) x^n = Sum_{k>0} x^a(k). - Michael Somos, Oct 21 2006
a(A095114(n)) = n and a(m) < n for m < A095114(n). - Reinhard Zumkeller, Feb 09 2012 [First inequality corrected from a(m) < m by Glen Whitney, Oct 06 2015]
Conjecture: a(n) >= n^(phi-1) for all n. - Jianing Song, Aug 19 2021
a(n) = A095114(n+1) - A095114(n). - Alan Michael Gómez Calderón, Dec 21 2024 after Ralf Stephan

A006753 Smith (or joke) numbers: composite numbers k such that sum of digits of k = sum of digits of prime factors of k (counted with multiplicity).

Original entry on oeis.org

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985, 1086, 1111, 1165, 1219

Offset: 1

N. J. A. Sloane

Of course primes also have this property, trivially.
a(133809) = 4937775 is the first Smith number historically: 4937775 = 3*5*5*65837 and 4+9+3+7+7+7+5 = 3+5+5+(6+5+8+3+7) = 42, Albert Wilansky coined the term Smith number when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.
There are 248483 7-digit Smith numbers, corresponding to US phone numbers without area codes (like 4937775). - Charles R Greathouse IV, May 19 2013
A007953(a(n)) = Sum_{k=1..A001222(a(n))} A007953(A027746(a(n),k)), and A066247(a(n))=1. - Reinhard Zumkeller, Dec 19 2011
3^3, 3^6, 3^9, 3^27 are in the sequence. - Sergey Pavlov, Apr 01 2017
As mentioned by Giovanni Resta, there are no other terms of the form 3^t for 0 < t < 300000 and, probably, no other terms of such form for t >= 300000. It seems that, if there exists any other term of form 3^t with integer t, then t == 0 (mod 3) or, perhaps, t = {3^k; 2*3^k} where k is an integer, k > 10. - Sergey Pavlov, Apr 03 2017

			58 = 2*29; sum of digits of 58 is 13, sum of digits of 2 + sum of digits of 29 = 2+11 is also 13.

M. Gardner, Penrose Tiles to Trapdoor Ciphers. Freeman, NY, 1989, p. 300.
R. K. Guy, Unsolved Problems in the Theory of Numbers, Section B49.
C. A. Pickover, "A Brief History of Smith Numbers" in "Wonders of Numbers: Adventures in Mathematics, Mind and Meaning", pp. 247-248, Oxford University Press, 2000.
J. E. Roberts, Lure of the Integers, pp. 269-270 MAA 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. D. Spencer, Key Dates in Number Theory History, Camelot Pub. Co. FL, 1995, pp. 94.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.1.14 and 3.1.16 on pages 84-85.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 180.

T. D. Noe, Table of n, a(n) for n = 1..10000
K. S. Brown's Mathpages, Smith Numbers and Rhonda Numbers
C. K. Caldwell, The Prime Glossary, Smith number
P. J. Costello, Smith Numbers
M. Gardner, Letter to N. J. A. Sloane, Jun 20 1991.
Ely Golden, General program for generating Smith number sequences
S. S. Gupta, Smith Numbers
T. Jason, Smith number
Madras Math's Amazing Number Facts, Smith Numbers.
Wayne L. McDaniel, The Existence of Infinitely Many k-Smith Numbers, The Fibonacci Quarterly, 25(1), 76-80, (1987).
Sham Oltikar, and Keith Wayland, Construction of Smith Numbers, Mathematics Magazine, vol. 56(1), 1983, pp. 36-37.
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
Carlos Rivera, Problem 107: Consecutive Smith numbers, The Prime Puzzles and Problems Connection.
Carlos Rivera, Problem 108: Methods for generating Smith numbers, The Prime Puzzles and Problems Connection.
W. Schneider, Smith Numbers
Eric Weisstein's World of Mathematics, Smith Number
Wikipedia, Smith number
A. Wilansky, Smith numbers, Two-Year Coll. Math. J., 13 (1982), p. 21.
A. Witno, A Family of Sequences Generating Smith Numbers, J. Int. Seq. 16 (2013) #13.4.6

Cf. A002808, A007953, A019506, A050218, A050224, A050255, A098834-A098840, A103123-A103126, A104166-A104171, A104390, A104391, A202387, A202388.

a006753 n = a006753_list !! (n-1)
a006753_list = [x | x <- a002808_list,
                    a007953 x == sum (map a007953 (a027746_row x))]
-- Reinhard Zumkeller, Dec 19 2011

q:= n-> not isprime(n) and (s-> s(n)=add(s(i[1])*i[2], i=
     ifactors(n)[2]))(h-> add(i, i=convert(h, base, 10))):
select(q, [$1..2000])[];  # Alois P. Heinz, Apr 22 2021

fQ[n_] := !PrimeQ@ n && n>1 && Plus @@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]] }] & /@ FactorInteger@ n] == Plus @@ IntegerDigits@ n; Select[ Range@ 1200, fQ]

isA006753(n) = if(isprime(n), 0, my(f=factor(n)); sum(i=1,#f[,1], sumdigits(f[i,1])*f[i,2]) == sumdigits(n)); \\ Charles R Greathouse IV, Jan 03 2012; updated by Max Alekseyev, Oct 21 2016

from sympy import factorint
def sd(n): return sum(map(int, str(n)))
def ok(n):
  f = factorint(n)
  return sum(f[p] for p in f) > 1 and sd(n) == sum(sd(p)*f[p] for p in f)
print(list(filter(ok, range(1220)))) # Michael S. Branicky, Apr 22 2021

is_A006753 = lambda n: n > 1 and not is_prime(n) and sum(n.digits()) == sum(sum(p.digits())*m for p,m in factor(n)) # D. S. McNeil, Dec 28 2010

A025748 3rd-order Patalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260, 156636118477018620, 1324287183487521060

Offset: 0

Olivier Gérard

G.f. (with a(0)=0) is series reversion of x - 3*x^2 + 3*x^3.

The Hankel transform of a(n) is A005130(n) * 3^binomial(n,2).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
I. M. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005, eq. (5.1).
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.

Apart from the initial 1, identical to A097188.

Cf. A004989, A005130, A034000, A034164, A073006,

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (4 - (1-9*x)^(1/3))/3 )); // G. C. Greubel, Sep 17 2019

A025748 :=proc(n)
        local x;
        coeftayl(4-(1-9*x)^(1/3),x=0,n) ;
        %/3 ;
end proc: # R. J. Mathar, Nov 01 2012

CoefficientList[Series[(4-Power[1-9x, (3)^-1])/3,{x,0,25}],x] (* Harvey P. Dale, Nov 14 2011 *)
Flatten[{1,Table[FullSimplify[9^(n-1) * Gamma[n-1/3] / (n * Gamma[2/3] * Gamma[n])],{n,1,25}]}] (* Vaclav Kotesovec, Feb 09 2014 *)
a[n_] := 9^(n-1) * Pochhammer[2/3, n-1]/n!; a[0] = 1; Array[a, 25, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n)=if(n<1,n==0,polcoeff(serreverse(x-3*x^2+3*x^3+x*O(x^n)),n))

def A025748_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((4 - (1-9*x)^(1/3))/3).list()
A025748_list(25) # G. C. Greubel, Sep 17 2019

From Wolfdieter Lang: (Start)
G.f.: (4 - (1-9*x)^(1/3))/3.
a(n) = 3^(n-1) * 2 * A034000(n-1)/n!, n >= 2.
a(n) = 3 * A034164(n-2), n >= 2. (End)
D-finite with recurrence n*a(n) + 3*(4-3*n)*a(n-1) = 0, n >= 2. - R. J. Mathar, Oct 29 2012
For n>0, a(n) = 9^(n-1) * Gamma(n-1/3) / (n * Gamma(2/3) * Gamma(n)). - Vaclav Kotesovec, Feb 09 2014
For n > 0, a(n) = 3^(2*n-1)*(-1)^(n+1)*binomial(1/3, n). - Peter Bala, Mar 01 2022
Sum_{n>=0} 1/a(n) = 37/16 + 3*sqrt(3)*Pi/64 - 9*log(3)/64. - Amiram Eldar, Dec 02 2022
For n >= 1, a(n) = Integral_{x = 0..9} x^n * w(x) dx, where w(x) = 1/(2*sqrt(3)*Pi) *  x^(2/3)*(9 - x)^(1/3)/x^2. - Peter Bala, Oct 14 2024
a(n) ~ 9^(n-1) / (Gamma(2/3) * n^(4/3)). - Amiram Eldar, Aug 20 2025

A005156 Number of alternating sign 2n+1 X 2n+1 matrices symmetric about the vertical axis (VSASM's); also 2n X 2n off-diagonally symmetric alternating sign matrices (OSASM's).

Original entry on oeis.org

1, 1, 3, 26, 646, 45885, 9304650, 5382618660, 8878734657276, 41748486581283118, 559463042542694360707, 21363742267675013243931852, 2324392978926652820310084179576, 720494439459132215692530771292602232, 636225819409712640497085074811372777428304

Offset: 0

N. J. A. Sloane

a(n+1) is the Hankel transform of A006013. - Paul Barry, Jan 20 2007

a(n+1) is the Hankel transform of A025174(n+1). - Paul Barry, Apr 14 2008

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 201, VS(2n+1).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gheorghe Coserea, Table of n, a(n) for n = 0..66
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 23.
M. T. Batchelor, J. de Gier and B. Nienhuis, The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions, arXiv:cond-mat/0101385 [cond-mat.stat-mech], 2001, (see A_V(2n+1)).
N. T. Cameron, Random walks, trees and extensions of Riordan group techniques, Dissertation, Howard University, 2002.
J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.
I. Fischer, The number of monotone triangles with prescribed bottom row, arXiv:math/0501102 [math.CO], 2005.
I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.
W. Hebsich and M. Rubey, Extended Rate, More Gfun, arXiv:math/0702086 [math.CO], 2007. [See p. 23.]
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001, (see A_V(2n+1)).
A. V. Razumov and Yu. G. Stroganov, On refined enumerations of some symmetry classes of alternating sign matrices, arXiv:math-ph/0312071, 2003.
D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math/0008045 [math.CO], 2000.
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

Cf. A109074, A134357.

A005156 := proc(n) local i,j,t1; (-3)^(n^2)*mul( mul( (6*j-3*i+1)/(2*j-i+2*n+1), j=1..n ),i=1..2*n+1); end;

Table[1/2^n Product[((6k-2)!(2k-1)!)/((4k-1)!(4k-2)!),{k,n}],{n,0,20}] (* Harvey P. Dale, Jul 07 2011 *)

a(n) = prod(k = 0, n-1, (3*k+2)*(6*k+3)!*(2*k+1)!/((4*k+2)!*(4*k+3)!));
vector(15, n, a(n-1))  \\ Gheorghe Coserea, May 30 2016

The formula for a(n) (see the Maple code) was conjectured by Robbins and proved by Kuperberg.
a(n) = (1/2^n) * Product_{k=1..n} ((6k-2)!(2k-1)!)/((4k-1)!(4k-2)!) (Razumov/Stroganov).
a(n) ~ exp(1/72) * Pi^(1/6) * 3^(3*n^2 + 3*n/2 + 11/72) / (A^(1/6) * GAMMA(1/3)^(1/3) * n^(5/72) * 2^(4*n^2 + 3*n + 1/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

cp's OEIS Frontend

A049503 a(n) = A005130(n)^2.

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A160708 Convolution triangle by rows, row sums = the Robbins sequence, A005130 starting with offset 1.

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A194827 2-adic valuation of the number of n X n Alternating Sign Matrices (A005130(n)).

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A227833 3-adic valuation of A005130(n).

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A160707 Convolution square root of the Robbins sequence, A005130, starting with offset 1.

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A173312 Partial sums of A005130.

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A001462 Golomb's sequence: a(n) is the number of times n occurs, starting with a(1) = 1.

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