A153881 -id:A153881 - cp's OEIS frontend

A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 26, 36, 12, 1, 0, 150, 250, 120, 20, 1, 0, 1082, 2040, 1230, 300, 30, 1, 0, 9366, 19334, 13650, 4270, 630, 42, 1, 0, 94586, 209580, 166376, 62160, 11900, 1176, 56, 1, 0, 1091670, 2562354, 2229444, 952728, 220500, 28476, 2016, 72, 1

Offset: 0

Wolfdieter Lang, May 04 2007

Matrix product of Stirling2 with unsigned Stirling1 triangle.
For the subtriangle without column no. m=0 and row no. n=0 see A079641.
The reversed matrix product |S1|. S2 is given in A111596.
As a product of lower triangular Jabotinsky matrices this is a lower triangular Jabotinsky matrix. See the D. E. Knuth references given in A039692 for Jabotinsky type matrices.
E.g.f. for row polynomials P(n,x):=sum(a(n,m)*x^m,m=0..n) is 1/(2-exp(z))^x. See the e.g.f. for the columns given below.
A048993*A132393 as infinite lower triangular matrices. - Philippe Deléham, Nov 01 2009
Triangle T(n,k), read by rows, given by (0,2,1,4,2,6,3,8,4,10,5,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 19 2011.
Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle begins:
  1;
  0,    1;
  0,    2,    1;
  0,    6,    6,    1;
  0,   26,   36,   12,   1;
  0,  150,  250,  120,  20,  1;
  0, 1082, 2040, 1230, 300, 30,  1;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Olivier Bodini, Antoine Genitrini, Cécile Mailler, Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Wolfdieter Lang, First ten rows and more.

Columns k=0..3 give A000007, A000629(n-1), A129063, A129064.

Cf. A000629, A000670, A005649, A079641, A129062, A325872, A325873, A383140.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> polylog(-n,1/2), 9); # Peter Luschny, Jan 27 2016

rows = 9;
t = Table[PolyLog[-n, 1/2], {n, 0, rows}]; T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)
p[n_] := Sum[StirlingS2[n, k] Pochhammer[x, k], {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten (* Peter Luschny, Jun 27 2019 *)

def a_row(n):
    s = sum(stirling_number2(n,k)*rising_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)] # Peter Luschny, Jun 28 2019

a(n,m) = Sum_{k=m..n} S2(n,k) * |S1(k,m)|, n>=0; S2=A048993, S1=A048994.
E.g.f. of column k (with leading zeros): (f(x)^k)/k! with f(x):= -log(1-(exp(x)-1)) = -log(2-exp(x)).
Sum_{0<=k<=n} T(n,k)*x^k = A153881(n+1), A000007(n), A000670(n), A005649(n) for x = -1,0,1,2 respectively. - Philippe Deléham, Nov 19 2011

New name by Peter Luschny, Jun 27 2019

A050354 Number of ordered factorizations of n with one level of parentheses.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 9, 3, 5, 1, 21, 1, 5, 5, 27, 1, 21, 1, 21, 5, 5, 1, 81, 3, 5, 9, 21, 1, 37, 1, 81, 5, 5, 5, 111, 1, 5, 5, 81, 1, 37, 1, 21, 21, 5, 1, 297, 3, 21, 5, 21, 1, 81, 5, 81, 5, 5, 1, 201, 1, 5, 21, 243, 5, 37, 1, 21, 5, 37, 1, 513, 1, 5, 21, 21, 5, 37, 1, 297, 27, 5, 1, 201

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

Dirichlet inverse of (A074206*A153881). - Mats Granvik, Jan 12 2009

			For n=6, we have (6) = (3*2) = (2*3) = (3)*(2) = (2)*(3), thus a(6) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A002033, A050351, A050352, A050353, A050355, A050356, A050357, A050358, A050359.

A[n_]:=If[n==1, n/2, 2*Sum[If[dIndranil Ghosh, May 19 2017 *)

A050354aux(n) = if(1==n,n/2, 2*sumdiv(n,d, if(dA050354aux(d), 0)));
A050354(n) = if(1==n,n,A050354aux(n)); \\ Antti Karttunen, May 19 2017, after Jovovic's general recurrence.

def A(n): return 1/2 if n==1 else 2*sum(A(d) for d in divisors(n) if dIndranil Ghosh, May 19 2017, after Antti Karttunen's PARI program

Dirichlet g.f.: (2-zeta(s))/(3-2*zeta(s)).
Recurrence for number of ordered factorizations of n with k-1 levels of parentheses is a(n) = k*Sum_{d|n, d1, a(1)= 1/k. - Vladeta Jovovic, May 25 2005
a(p^k) = 3^(k-1).
a(A002110(n)) = A050351(n).
Sum_{k=1..n} a(k) ~ -n^r / (4*r*Zeta'(r)), where r = 2.185285451787482231198145140899733642292971552057774261555354324536... is the root of the equation Zeta(r) = 3/2. - Vaclav Kotesovec, Feb 02 2019

Duplicate comment removed by R. J. Mathar, Jul 15 2010

A154990 Triangle read by rows. Main diagonal is positive. The rest of the terms are negative.

Original entry on oeis.org

1, -1, 1, -1, -1, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1

Offset: 1

Mats Granvik, Jan 18 2009

Triangle can be used in matrix inverses. Signs in columns as in A153881.

Iff n is a triangular number, a(n)=1; otherwise, a(n)=-1. (This is explicitly implemented in the second Mathematica program below.) - Harvey P. Dale, Apr 27 2014

			Table begins:
   1;
  -1,  1;
  -1, -1,  1;
  -1, -1, -1,  1;
  -1, -1, -1, -1,  1;
  -1, -1, -1, -1, -1,  1;
  -1, -1, -1, -1, -1, -1, 1;

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Cf. A023443, A023444.

[k eq n select 1 else -1: k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 06 2021

A154990 := proc(n,k)
    option remember;
    if k = n then
        1;
    elif k > n then
        0;
    else
        -1 ;
    end if;
end proc:
seq(seq(A154990(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Sep 16 2017

Flatten[Table[PadLeft[{1},n,-1],{n,15}]] (* or *) With[{tr=Accumulate[ Range[ 15]]}, Table[If[MemberQ[tr,n],1,-1],{n,Last[tr]}]] (* Harvey P. Dale, Apr 27 2014 *)

flatten([[1 if k==n else -1 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 06 2021

From G. C. Greubel, Mar 06 2021: (Start)
T(n, k) = -1 with T(n, n) = 1.
Sum_{k=1..n} T(n, k) = 2-n = -A023443(n-1) = -A023444(n). (End)

A165047 Consider the base-5 Kaprekar map x->K(x) described in A165032. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 48, 45781056, 1992, 7488, 249992, 26648194761946797370910644531056, 170312312, 447082519531056, 953674316406249992, 43487548828124832, 68219378590583801269531056

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 100 base-5 digits):
a(1) = 0 (base 10) = 0 (base 5)
a(2) = 48 (base 10) = 143 (base 5)
a(3) = 45781056 (base 10) = 43204443211 (base 5)
a(4) = 1992 (base 10) = 30432 (base 5)
a(5) = 7488 (base 10) = 214423 (base 5)
a(6) = 249992 (base 10) = 30444432 (base 5)
a(7) = 26648194761946797370910644531056 (base 10) = 432044444444444444444444444444444444444443211 (base 5)
a(8) = 170312312 (base 10) = 322044443222 (base 5)
a(9) = 447082519531056 (base 10) = 432044444444444443211 (base 5)
a(10) = 953674316406249992 (base 10) = 30444444444444444444444432 (base 5)
a(11) = 43487548828124832 (base 10) = 331044444444444444443312 (base 5)
a(12) = 68219378590583801269531056 (base 10) = 4320444444444444444444444444444443211 (base 5)
a(13) = 388774887899923005107893914100714027881532907485961914056 (base 10) = 432222222222222222222222044444444444444444444444444444443222222222222222222222211 (base 5)
a(14) = 4366040229797363281056 (base 10) = 4320444444444444444444444443211 (base 5)
a(15) = 15550995515996920287582474884401599410921335220336914056 (base 10) = 4322222222222222222222222222222222044444444432222222222222222222222222222222211 (base 5)
a(18) = 1705484464764595031738281056 (base 10) = 432044444444444444444444444444444443211 (base 5)
a(20) = 6505906924303417326882481575012207031056 (base 10) = 432044444444444444444444444444444444444444444444444443211 (base 5)
a(21) = 416378043155418708920478820800781056 (base 10) = 432044444444444444444444444444444444444444444443211 (base 5)
a(22) = 39708904567281599895522958831861615180969238281056 (base 10) = 43204444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(23) = 16655121726216748356819152832031056 (base 10) = 4320444444444444444444444444444444444444444443211 (base 5)
a(24) = 1479271969470441337508073405618280737883196707116439938545227050781056 (base 10) = 432044444444444444444444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(26) = 260236276972136693075299263000488281056 (base 10) = 4320444444444444444444444444444444444444444444444443211 (base 5)
a(27) = 9694556779121484349492909871059964643791317939758300781056 (base 10) = 43204444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(28) = 151477449673773192960826716735311947559239342808723449707031056 (base 10) = 43204444444444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(29) = 4066191827689635829301550984382629394531056 (base 10) = 4320444444444444444444444444444444444444444444444444444443211 (base 5)
a(30) = 101654795692240895732538774609565734863281056 (base 10) = 432044444444444444444444444444444444444444444444444444444443211 (base 5)
a(33) = 1588356182691263995820918353274464607238769531056 (base 10) = 432044444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(35) = 992722614182039997388073970796540379524230957031056 (base 10) = 4320444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(36) = 59170878778817653500322936224731229515327868284657597541809082031056 (base 10) = 4320444444444444444444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(39) = 387782271164859373979716394842398585751652717590332031056 (base 10) = 432044444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(41) = 242363919478037108737322746776499116094782948493957519531056 (base 10) = 4320444444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)

Joseph Myers, Table of n, a(n) for n=1..15
Index entries for the Kaprekar map

Cf. A165032, A165036, A165037, A165039, A165041, A165043.

In other bases: A153881 (base 2), A165008 (base 3), A165028 (base 4), A165067 (base 6), A165086 (base 7), A165106 (base 8), A165126 (base 9), A151959 (base 10).

A165126 Consider the base-9 Kaprekar map x->K(x) described in A165110. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 16, 2256, 31596672, 34960, 26531651360, 14560721001508880, 8724454714749973651840, 108401672318914272, 711223428647787942432, 16513410921312, 278474880, 4754966263206652084045296, 183696

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 70 base-9 digits):
a(1) = 0 (base 10) = 0 (base 9)
a(2) = 16 (base 10) = 17 (base 9)
a(3) = 2256 (base 10) = 3076 (base 9)
a(4) = 31596672 (base 10) = 65407433 (base 9)
a(5) = 34960 (base 10) = 52854 (base 9)
a(6) = 26531651360 (base 10) = 75430875432 (base 9)
a(7) = 14560721001508880 (base 10) = 77643208887654212 (base 9)
a(8) = 8724454714749973651840 (base 10) = 87654320888888876543211 (base 9)
a(9) = 108401672318914272 (base 10) = 644444418864444443 (base 9)
a(10) = 711223428647787942432 (base 10) = 6444444441886444444443 (base 9)
a(11) = 16513410921312 (base 10) = 64418888886443 (base 9)
a(12) = 278474880 (base 10) = 641888643 (base 9)
a(13) = 4754966263206652084045296 (base 10) = 65544444218888886644444333 (base 9)
a(14) = 183696 (base 10) = 308876 (base 9)
a(15) = 8780535458788649952 (base 10) = 64444441888864444443 (base 9)
a(16) = 8811048483031324779456676539593726674416 (base 10) = 655444442188888888888888888888886644444333 (base 9)
a(18) = 177097392902234856396140027020301600 (base 10) = 7766644432208888888888888766544422212 (base 9)
a(20) = 50771339309018227821951440 (base 10) = 776444432088888887654444212 (base 9)
a(21) = 124998824875093374012011515622478472976 (base 10) = 7544444444444421888888886644444444444432 (base 9)
a(22) = 31197333902107825741164471552 (base 10) = 655444444444308875444444444333 (base 9)
a(23) = 685322163857921701893212141347733334983278765142051291692546183840 (base 10) = 876544444444444432088888888888888888888888888888887654444444444443211 (base 9)
a(25) = 702826002884083319045760971727413317645857409225018391041723392 (base 10) = 655444444444444444443088888888888888888888887544444444444444444333 (base 9)
a(27) = 94116815581356594318021072974737787790560 (base 10) = 7766644432208888888888888888888766544422212 (base 9)
a(28) = 2156904722606587695378609845389399883833898563216184240 (base 10) = 777777655554444333332222210888776666655555444433332111112 (base 9)
a(29) = 26981972242036651232570366433200 (base 10) = 776444444443208888876544444444212 (base 9)
a(31) = 377773874412068206712875872 (base 10) = 6441888888888888888888886443 (base 9)
a(35) = 2520442200659768347220271484032 (base 10) = 65408888888888888888888888887433 (base 9)
a(39) = 1161485426793562822354375723686123973520 (base 10) = 77644444320888888888888888888876544444212 (base 9)
a(40) = 16327050854444484146838503988925985281714578428892247536 (base 10) = 6554444444444444444444442188888866444444444444444444444333 (base 9)
a(46) = 10150412679066664692018845810715370585139195588212689167959856 (base 10) = 7665444444444444444444444422218888666644444444444444444444443222 (base 9)
a(65) = 26152889553714885216926446303415314330076524215805520 (base 10) = 7654308888888888888888888888888888888888888888888754322 (base 9)

Joseph Myers, Table of n, a(n) for n=1..16
Index entries for the Kaprekar map

Cf. A165110, A165114, A165115, A165117, A165119, A165121.

In other bases: A153881 (base 2), A165008 (base 3), A165028 (base 4), A165047 (base 5), A165067 (base 6), A165086 (base 7), A165106 (base 8), A151959 (base 10).

A165008 Consider the base-3 Kaprekar map x->K(x) described in A164993. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 32, 320, 26240, 1240024, 11160256, 2297798771761759543384, 15075857741528904364175224, 8135830264, 5931020266096, 659002251784, 350220815692997944

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 200 base-3 digits):
a(1) = 0 (base 10) = 0 (base 3)
a(2) = 32 (base 10) = 1012 (base 3)
a(3) = 320 (base 10) = 102212 (base 3)
a(4) = 26240 (base 10) = 1022222212 (base 3)
a(5) = 1240024 (base 10) = 2022222222211 (base 3)
a(6) = 11160256 (base 10) = 202222222222211 (base 3)
a(7) = 2297798771761759543384 (base 10) = 202222222222222222222222222222222222222222211 (base 3)
a(8) = 15075857741528904364175224 (base 10) = 20222222222222222222222222222222222222222222222222211 (base 3)
a(9) = 8135830264 (base 10) = 202222222222222222211 (base 3)
a(10) = 5931020266096 (base 10) = 202222222222222222222222211 (base 3)
a(11) = 659002251784 (base 10) = 2022222222222222222222211 (base 3)
a(12) = 350220815692997944 (base 10) = 2022222222222222222222222222222222211 (base 3)
a(14) = 480412641554176 (base 10) = 2022222222222222222222222222211 (base 3)
a(15) = 1202547548374693105751742636119782969638250884664 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(18) = 3151987341236981536 (base 10) = 202222222222222222222222222222222222211 (base 3)
a(20) = 1221144477063841253498193544 (base 10) = 202222222222222222222222222222222222222222222222222222211 (base 3)
a(21) = 1675095304614322707130576 (base 10) = 202222222222222222222222222222222222222222222222211 (base 3)
a(22) = 5840696178317563736403001300866976 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(23) = 186121700512702523014504 (base 10) = 2022222222222222222222222222222222222222222222211 (base 3)
a(24) = 133616394263854789527971404013309218848694542736 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(26) = 135682719673760139277577056 (base 10) = 2022222222222222222222222222222222222222222222222222211 (base 3)
a(27) = 3103985417701264389637747414334049249616 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(28) = 2262805369504221740045917865049521902973704 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(29) = 98912702642171141533353677464 (base 10) = 2022222222222222222222222222222222222222222222222222222222211 (base 3)
a(30) = 890214323779540273800183097216 (base 10) = 202222222222222222222222222222222222222222222222222222222222211 (base 3)
a(33) = 648966242035284859600333477874104 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(34) = 1624463421305399727955317383718278234275809671395357663852488383536 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(35) = 52566265604858073627627011707802824 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(36) = 14846266029317198836441267112589913205410504744 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(39) = 344887268633473821070860823814894361064 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(40) = 1166199021234168600322186441674903634837253284669268066647739969989548815584492779042071704 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(41) = 27935868759311379506739726729006443246584 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(42) = 95922940564662548536033536191180611455752285286224474692825586559712504 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(44) = 51765728804119417583918760520852820074569256760349620344 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(46) = 131581537125737377964380708081180536976340583383023970772051559066816 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(48) = 850159086479708909634873915981004749796357644523896420586202438122381086561095235921670275856 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(50) = 10822927935372237951765683725078046726744257962016 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(51) = 97406351418350141565891153525702420540698321658184 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(52) = 50977383456624829456538198506977215332656450244796421055229922546880175495464 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(53) = 7889914464886361466837183435581896063796564054313304 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(55) = 37737216298203055418676776419701705834360988178294873234416 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(60) = 1184233834131636401679426372730624832787065250447215736948464031601384 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(65) = 2228344885192592219417444970806966027813181990940133969619325624 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(69) = 14620170791748597551597856453464504108482287042558218974672395451864 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(74) = 863306465081962936824301825720625503101770567576020272235430279037412576 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(78) = 3010163515730239554579124083638497588178030730504983866890271696470727482831948976 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(81) = 4129168059986611185979594079065154441945172469828510105473623726297294215132984 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(83) = 334462612858915506064347120404277509797558970056109318543363521830080831425772104 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(86) = 243823244774149403920909050774718304642420489170903693218112007414128926109387867456 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(89) = 177747145440354915458342698014769644084324536605588792356003653404899987133743755379064 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(90) = 1599724308963194239125084282132926796758920829450299131204032880644099884203693798411616 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(95) = 94462120719967656626097101775667194421817516058210713398466937569153454062343915102407808424 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(98) = 68862886004856421680424787194461384733504969206435610067482397487912868011448714109655292344736 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)

Index entries for the Kaprekar map

Cf. A164993, A164997, A164998, A165000, A165002, A165004.

In other bases: A153881 (base 2), A165028 (base 4), A165047 (base 5), A165067 (base 6), A165086 (base 7), A165106 (base 8), A165126 (base 9), A151959 (base 10).

A165028 Consider the base-4 Kaprekar map x->K(x) described in A165012. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 126, 41958

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 200 base-4 digits):
a(1) = 0 (base 10) = 0 (base 4)
a(2) = 126 (base 10) = 1332 (base 4)
a(3) = 41958 (base 10) = 22033212 (base 4)

Index entries for the Kaprekar map

Cf. A165012, A165016, A165017, A165019, A165021, A165023.

In other bases: A153881 (base 2), A165008 (base 3), A165047 (base 5), A165067 (base 6), A165086 (base 7), A165106 (base 8), A165126 (base 9), A151959 (base 10).

A165067 Consider the base-6 Kaprekar map x->K(x) described in A165051. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 4305, 16840

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 100 base-6 digits):
a(1) = 0 (base 10) = 0 (base 6)
a(2) = 4305 (base 10) = 31533 (base 6)
a(3) = 16840 (base 10) = 205544 (base 6)
a(6) = 430 (base 10) = 1554 (base 6)
a(7) = 895275 (base 10) = 31104443 (base 6)

Index entries for the Kaprekar map

Cf. A165051, A165055, A165056, A165058, A165060, A165062.

In other bases: A153881 (base 2), A165008 (base 3), A165028 (base 4), A165047 (base 5), A165086 (base 7), A165106 (base 8), A165126 (base 9), A151959 (base 10).

A165086 Consider the base-7 Kaprekar map x->K(x) described in A165071. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 144, 1068, 9458722410775248, 9936, 55500, 65945195409025452

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 100 base-7 digits):
a(1) = 0 (base 10) = 0 (base 7)
a(2) = 144 (base 10) = 264 (base 7)
a(3) = 1068 (base 10) = 3054 (base 7)
a(4) = 9458722410775248 (base 10) = 5544222066654442212 (base 7)
a(5) = 9936 (base 10) = 40653 (base 7)
a(6) = 55500 (base 10) = 320544 (base 7)
a(7) = 65945195409025452 (base 10) = 55332221066554443312 (base 7)
a(9) = 419850417612 (base 10) = 42222166444443 (base 7)
a(10) = 114965566537586468276798389479111631100827277423731225926928273344 (base 10) = 65444444444444444444444443066666666666666666666666532222222222222222222222211 (base 7)
a(11) = 31412208 (base 10) = 530666532 (base 7)
a(12) = 26884299308652 (base 10) = 5443216666443222 (base 7)
a(13) = 894060461610805641013834968 (base 10) = 54444444322106666665544322222222 (base 7)
a(14) = 1591271424672409468790707489057394638817384701224062547077367141620193382944 (base 10) = 65444444444444444444444444444306666666666666666666666666665322222222222222222222222222211 (base 7)
a(17) = 107837050564847832079804652808012 (base 10) = 55444444332221110666655554443322222212 (base 7)
a(24) = 7598644111289477155212 (base 10) = 54443222221066554444432222 (base 7)
a(25) = 18244344524504743400068812 (base 10) = 544432222222106655444444432222 (base 7)

Index entries for the Kaprekar map

Cf. A165071, A165075, A165076, A165078, A165080, A165082.

In other bases: A153881 (base 2), A165008 (base 3), A165028 (base 4), A165047 (base 5), A165067 (base 6), A165106 (base 8), A165126 (base 9), A151959 (base 10).

cp's OEIS Frontend

A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

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A050354 Number of ordered factorizations of n with one level of parentheses.

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A154990 Triangle read by rows. Main diagonal is positive. The rest of the terms are negative.

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A165047 Consider the base-5 Kaprekar map x->K(x) described in A165032. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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A165126 Consider the base-9 Kaprekar map x->K(x) described in A165110. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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A165008 Consider the base-3 Kaprekar map x->K(x) described in A164993. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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A165028 Consider the base-4 Kaprekar map x->K(x) described in A165012. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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A165067 Consider the base-6 Kaprekar map x->K(x) described in A165051. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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A165086 Consider the base-7 Kaprekar map x->K(x) described in A165071. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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