A035927 -id:A035927 - cp's OEIS frontend

A000582 a(n) = binomial coefficient C(n,9).

1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 92378, 167960, 293930, 497420, 817190, 1307504, 2042975, 3124550, 4686825, 6906900, 10015005, 14307150, 20160075, 28048800, 38567100, 52451256, 70607460, 94143280, 124403620, 163011640, 211915132

Offset: 9

N. J. A. Sloane

Figurate numbers based on 9-dimensional regular simplex. - Jonathan Vos Post, Nov 28 2004
Product of 9 consecutive numbers divided by 9!. - Artur Jasinski, Dec 02 2007
In this sequence there are no primes. - Artur Jasinski, Dec 02 2007
a(9+n) gives the number of words with n letters over the alphabet {0,1,..,9} such that these letters are read from left to right in weakly increasing (nondecreasing) order. - R. J. Cano, Jul 20 2014
a(n) = fallfac(n, 9)/9! = binomial(n, 9) is also the number of independent components of an antisymmetric tensor of rank 9 and dimension n >= 9 (for n=1..8 this becomes 0). Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
From Juergen Will, Jan 23 2016: (Start)
Number of compositions (ordered partitions) of n+1 into exactly 10 parts.
Number of weak compositions (ordered weak partitions) of n-9 into exactly 10 parts. (End)
Number of integers divisible by 9 in the interval [0, 10^(n-8)-1]. - Miquel Cerda, Jul 02 2017

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 9..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 259
Milan Janjic, Two Enumerative Functions
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Ch. Stover and E. W. Weisstein, Composition. From MathWorld - A Wolfram Web Resource.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A053138, A053131, A000581, A035927, A001787, A242091.

[Binomial(n,9) : n in [9..50]]; // Wesley Ivan Hurt, Jul 20 2014

A000582 := n->binomial(n,9): seq(A000582(n), n=9..40);
A000582:=1/(z-1)**10; # Simon Plouffe in his 1992 dissertation (offset 0)
seq(binomial(n,9),n=0..29); # Zerinvary Lajos, Jun 23 2008, R. J. Mathar, Jul 07 2009

Table[n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)/9!,{n,100}] (* Artur Jasinski, Dec 02 2007 *)
Table[Binomial[n, 9], {n, 9, 50}] (* Wesley Ivan Hurt, Jul 20 2014 *)

a(n)=binomial(n,9) \\ Charles R Greathouse IV, Jul 21 2014

G.f.: x^9/(1-x)^10.
a(n) = -A110555(n+1, 9). - Reinhard Zumkeller, Jul 27 2005
a(n+8) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)/9!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009
Sum_{k>=9} 1/a(k) = 9/8. - Tom Edgar, Sep 10 2015
Sum_{n>=9} (-1)^(n+1)/a(n) = A001787(9)*log(2) - A242091(9)/8! = 2304*log(2) - 446907/280 = 0.9146754386... - Amiram Eldar, Dec 10 2020

Formulas referring to other offsets rewritten by R. J. Mathar, Jul 07 2009

A179239 Permutation classes of integers, each identified by its smallest member.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 44, 45, 46, 47, 48, 49, 50, 55, 56, 57, 58, 59, 60, 66, 67, 68, 69, 70, 77, 78, 79, 80, 88, 89, 90, 99, 100, 101, 102, 103

Offset: 0

Aaron Dunigan AtLee, Jul 04 2010

Let the "permutation set" of a positive integer n be the set of all integers formed by permuting the digits of n. Two integers are "permutationally congruent" if they generate the same permutation set. A "permutation class" is a set of all permutationally congruent integers. This sequence lists each permutation class, identified by its smallest member.
These are also the positive integers in order, omitting any d-digit number n if a previously listed d-digit number is a permutation of the digits of n.
Range of A328447: smallest representative of the equivalence class of all numbers having the same digits up to permutation. Equivalently: Numbers with digits in nondecreasing order, except that the smallest nonzero digit must precede the zero digits. This sequence is useful when considering functions which depend only on the digits of n, e.g., the number of primes contained in n, cf. A039993, A039999, A075053 and the records therein, A072857 (primeval numbers) and A076497, resp. A239196 and A239197, etc. - M. F. Hasler, Oct 18 2019

			The permutation set of 24 is {24, 42}, and this is the equivalence class modulo permutations of both of them, so 24 is listed, but 42 is not.
The permutation set of 30 is {3, 30}, but 3 is not in the same permutation class as 30 since 30 cannot be obtained by permuting digits of 3. Therefore 30 is listed separately from 3.
The numbers 89 and 98 are also permutationally congruent and form a permutation class, so only the smaller one is listed.

Aaron Dunigan AtLee and Michael De Vlieger, Table of n, a(n) for n = 0..10000 (first 2997 terms from Aaron Dunigan AtLee; prefix 0 by _Georg Fischer_, Oct 24 2019)

A variant of A009994.
Cf. A047726, A035927 (Number of distinct n-digit numbers up to permutations of digits).
Cf. A004186, A328447: largest & smallest representative of the class of n.

maxTerm = 103; (*maxTerm is the greatest term you wish to see*) permutationSet[n_Integer] := FromDigits /@ Permutations[IntegerDigits[n]]; permutationCongruentQ[x_Integer, y_Integer] := Sort[permutationSet[x]] == Sort[permutationSet[y]]; DeleteDuplicates[Range[maxTerm], permutationCongruentQ]
f[n_] := Block[{a = {0}, b = {DigitCount[0]}, i, w}, Do[w = DigitCount@ i; AppendTo[b, w]; If[! MemberQ[Most@ b, w], AppendTo[a, i]], {i, n}]; Rest@ a]; f@ 103 (* or faster: *)
Select[Range@ 103, LessEqual @@ IntegerDigits@ # || And[Take[IntegerDigits@ #, Last@ DigitCount@ # + 1] == Reverse@ Take[Sort@ IntegerDigits@ #, Last@ DigitCount@ # + 1], LessEqual @@ DeleteCases[IntegerDigits@ #, d_ /; d == 0]] &] (* Michael De Vlieger, Jul 14 2015 *)

is(n) = {my(d=digits(n),i); for(i=2,#d, if(d[i]!=0, d=vecextract(d,concat([1],vector(#d-i+1,j,i-1+j))); break));d==vecsort(d)||n/10^valuation(n,10)<10}
\\given an element n, in base b, find the next element from the sequence.
nxt(n,{b=10}) = {my(d = digits(n)); i = #d; while(i>0&&d[i]==b-1,i--); if(i>1, if(d[i]>0, d[i]++, d[i]=d[1];);for(j=i+1,#d,d[j]=d[i]), if(i==1, d[i]++;for(j=2,#d,d[j]=0), return(10^(#d))));sum(j=1,#d,d[j]*10^(#d-j))} \\ David A. Corneth, Apr 23 2016

select( is_A179239(n)={n==A328447(n)}, [0..200]) \\ M. F. Hasler, Oct 18 2019

from itertools import count, chain, islice
from sympy.utilities.iterables import combinations_with_replacement
def A179239_gen(): # generator of terms
    return chain((0,),(int(a+''.join(b)) for l in count(1) for a in '123456789' for b in combinations_with_replacement('0'+''.join(str(d) for d in range(int(a),10)),l-1)))
A179239_list = list(islice(A179239_gen(),31)) # Chai Wah Wu, Sep 13 2022

Prefixed with a(0) = 0 by M. F. Hasler, Oct 18 2019

A014430 Subtract 1 from Pascal's triangle, read by rows.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 9, 4, 5, 14, 19, 14, 5, 6, 20, 34, 34, 20, 6, 7, 27, 55, 69, 55, 27, 7, 8, 35, 83, 125, 125, 83, 35, 8, 9, 44, 119, 209, 251, 209, 119, 44, 9, 10, 54, 164, 329, 461, 461, 329, 164, 54, 10, 11, 65, 219, 494, 791, 923, 791, 494, 219, 65, 11

Offset: 0

N. J. A. Sloane

Each value of the sequence (T(x,y)) is equal to the sum of all values in Pascal's Triangle that are in the rectangle defined by the tip (0,0) and the position (x,y). - Florian Kleedorfer (florian.kleedorfer(AT)austria.fm), May 23 2005
To clarify T(n,k) and A129696: We subtract I = Identity matrix from Pascal's triangle to obtain the beheaded variant, A074909. Then take column sums starting from the top of A074909 to get triangle A014430. Row sums of the inverse of triangle T(n,k) gives the Bernoulli numbers, A027641/A026642. Alternatively, triangle T(n,k) as an infinite lower triangular matrix * [the Bernoulli numbers as a vector] = [1, 1, 1, ...]. Given the B_n version starting (1, 1/2, 1/6, ...) triangle T(n,k) * the B_n vector [1, 1/2, 1/6, 0, -1/30, ...] = the triangular numbers. - Gary W. Adamson, Mar 13 2012
From R. J. Mathar, Apr 25 2016: (Start)
If regarded as a symmetric array of the form
  1  2   3   4    5  ...
  2  5   9  14   20  ...
  3  9  19  34   55  ...
  4 14  34  69  125  ...
  5 20  55 125  251  ...
  6 27  83 209  461  ...
  7 35 119 329  791  ...
  8 44 164 494 1286  ...
  9 54 219 714 2001  ...
it contains the rows (and columns) A000096, A062748, A063258, A062988, A124089, ..., A035927 and so on and counts the multisets of digits of numbers in base b>=2 with d>=1 digits (equivalent to the comment in A035927). (End)
Proof of Florian Kleedorfer's formula:  Take sums of the columns of the rectangle - these are all binomial coefficients by the Hockey Stick Identity. Note the locations of these coefficients:  They form a row going almost all the way to the edge, only missing the 1 - apply the Hockey Stick Identity again. - James East, Jul 03 2020

			Triangle begins:
  1;
  2,  2;
  3,  5,  3;
  4,  9,  9,   4;
  5, 14, 19,  14,   5;
  6, 20, 34,  34,  20,  6;
  7, 27, 55,  69,  55, 27,  7;
  8, 35, 83, 125, 125, 83, 35, 8;

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened

Cf. A000096, A007318, A014430, A026642, A027641, A027642, A035927.
Cf. A062748, A063258, A062988, A074909, A124089, A124326, A129696.
Triangle with zeros: A014473.
Cf. A000295 (row sums).

a014430 n k = a014430_tabl !! n !! k
a014430_row n = a014430_tabl !! n
a014430_tabl = map (init . tail) $ drop 2 a014473_tabl
-- Reinhard Zumkeller, Apr 10 2012

[Binomial(n+2,k+1)-1: k in [0..n], n in [0..13]]; // G. C. Greubel, Feb 25 2023

Table[Sum[Sum[Binomial[m, j], {m, j, j+(n-k)}], {j,0,k}], {n,0,10}, {k, 0,n}]//Flatten (* Michael De Vlieger, Sep 01 2020 *)
Table[Binomial[n+2,k+1] -1, {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 25 2023 *)

flatten([[binomial(n+2,k+1)-1 for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Feb 25 2023

T(n, k) = T(n-1, k) + T(n-1, k-1) + 1, T(0, 0)=1. - Ralf Stephan, Jan 23 2005
G.f.: 1 / ((1-x)*(1-x*y)*(1-x*(1+y))). - Ralf Stephan, Jan 24 2005
T(n, k) = Sum_{j=0..k} Sum_{m=j..j+(n-k)} binomial(m, j). - Florian Kleedorfer (florian.kleedorfer(AT)austria.fm), May 23 2005
T(n, k) = binomial(n+2, k+1) - 1. - G. C. Greubel, Feb 25 2023

More terms from Erich Friedman

Offset fixed by Reinhard Zumkeller, Apr 10 2012

A014553 Maximal multiplicative persistence (or length) of any n-digit number.

Original entry on oeis.org

1, 4, 5, 6, 7, 7, 8, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Eric W. Weisstein

The "persistence" or "length" of an N-digit decimal number is the number of times one must iteratively form the product of its digits until one obtains a one-digit product (For another definition see A003001.)

For all other n<2530, a(n)=11 because sequence is nondecreasing and a number with multiplicative persistence 12 must have more than 2530 digits. - Sascha Kurz, Mar 24 2002

			168889 is not in A003001 because a(6) = a(5) = 7.

Gottlieb, A. J. Problems 28-29 in "Bridge, Group Theory and a Jigsaw Puzzle." Techn. Rev. 72, unpaginated, Dec. 1969.
Gottlieb, A. J. Problem 29 in "Integral Solutions, Ladders and Pentagons." Techn. Rev. 72, unpaginated, Apr. 1970.

Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM, ITEM 56
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence.

Cf. A003001, A031346, A035927.

Corrected by N. J. A. Sloane, Nov 1995

More terms from John W. Layman, Mar 19 2002

A289410 Irregular triangular array T(m,k) with m (row) >= 1 and k (column) >= 1 read by rows: number of m-digit numbers whose digit sum is k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 54, 61, 66, 69, 70, 69, 66, 61, 54, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 219, 279, 342, 405, 465, 519, 564, 597, 615, 615, 597, 564, 519, 465, 405, 342, 279, 219, 165, 120, 84

Offset: 1

Miquel Cerda, Jul 05 2017

The m-th row is palindromic; T(m,k) = T(m,9*m+1-k).

			The irregular triangle T(m,k) begins:
m\k  1  2  3  4  5   6   7   8   9   10   11  12   13   14  15  16  17  18  19
1    1  1  1  1  1   1   1   1   1;
2    1  2  3  4  5   6   7   8   9    9    8   7    6    5   4   3   2   1;
3    1  3  6  10 15  21  28  36  45   54   61  66   69   70  69  66  61  54 45,...;
4    1  4  10 20 35  56  84  120 165  219  279 342  405  465,...;
5    1  5  15 35 70  126 210 330 495  714  992 1330 1725,...;
6    1  6  21 56 126 252 462 792 1287 2001 2992,...;
etc.
Row m(2), column k(4) there are 4 numbers of 2-digits whose digits sum = 4: 13, 22, 31, 40.

Miquel Cerda, Rows n=1..10 of triangle, flattened

The row sums = 9*10^(m-1) = A052268(n). The row lengths = 9*m = A008591(n). The middle diagonal = A071976. (row m=3) = A071817, (row m=4) = A090579, (row m=5) = A090580, (row m=6) = A090581, (row m=7) = A278969, (row m=8) = A278971, (row m=9) = A289354, (column k=3) = A000217, (column k=4) = A000292, (column k=5) = A000332, (column k=6) = A000389, (column k=7) = A000579, (column k=8) = A000580, (column k=9) = A000581, (column k=10) = A035927.

row:= proc(m) local g; g:= normal((1 - x^10)^(m-1)*(x - x^10)/(1 - x)^m);
seq(coeff(g,x,j),j=1..9*m) end proc:
seq(row(k),k=1..5); # Robert Israel, Jul 19 2017

G.f. of row m: (1 - x^10)^(m-1)*(x - x^10)/(1 - x)^m.

G.f. as array: (1+x+x^2)*(1+x^3+x^6)*x*y/(1-y*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)). - Robert Israel, Jul 19 2017

Edited by Robert Israel, Jul 19 2017

A115205 a(n) = binomial(n, 9) + 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 11, 56, 221, 716, 2003, 5006, 11441, 24311, 48621, 92379, 167961, 293931, 497421, 817191, 1307505, 2042976, 3124551, 4686826, 6906901, 10015006, 14307151, 20160076, 28048801, 38567101, 52451257, 70607461, 94143281, 124403621

Offset: 0

Roger L. Bagula, Mar 03 2006

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A035927.

[Binomial(n, 9)+1: n in [0..50]]; // Vincenzo Librandi, Feb 05 2016

seq(binomial(n,9)+1, n=0..26); # Zerinvary Lajos, Jan 13 2007

Table[1 + Binomial[n,9], {n, 0, 20}] (* G. C. Greubel, Feb 05 2016 *)

a(n) = binomial(n, 9) + 1; \\ Altug Alkan, Feb 05 2016

G.f.: ((3*x^2-3*x+1)*(3*x^6-9*x^5+18*x^4-21*x^3+15*x^2-6*x+1))/(x-1)^10. [Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009]

a(n) = (1/9!)*(n+1)*(n^8 - 37*n^7 + 583*n^6 - 5119*n^5 + 27568*n^4 - 94852*n^3 + 212976*n^2 - 322560*n + 9!) = binomial(n,9)+1. - G. C. Greubel, Feb 05 2016

A165618 a(n) = binomial(n+8,8) - 1.

Original entry on oeis.org

0, 8, 44, 164, 494, 1286, 3002, 6434, 12869, 24309, 43757, 75581, 125969, 203489, 319769, 490313, 735470, 1081574, 1562274, 2220074, 3108104, 4292144, 5852924, 7888724, 10518299, 13884155, 18156203, 23535819, 30260339, 38608019, 48903491

Offset: 0

Enrique Pérez Herrero, Sep 22 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. Pérez Herrero, Binomial Matrix (I) partitions, Psychedelic Geometry Blogspot, 09/22/09
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000096, A000581, A062748, A063258, A062988, A124089, A124090, A035927.

Table[ -1 + Binomial[n + 8, 8], {n, 0, 30}]
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,8,44,164,494,1286,3002,6434,12869},40] (* Harvey P. Dale, Nov 18 2013 *)

vector(100,n,binomial(n+7,8)-1) \\ Charles R Greathouse IV, May 27 2011

a(n) = binomial(n+8,8) - 1 = A000581(n+8) - 1.
a(n) = Sum_{r=1..n} binomial(8,r)*binomial(n,r).
a(n) = n(n+9)(n^6 + 27n^5 + 303n^4 + 1809n^3 + 6168n^2 + 11772n + 12176)/40320.

Edited by Charles R Greathouse IV, May 27 2011

cp's OEIS Frontend

A000582 a(n) = binomial coefficient C(n,9).

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A179239 Permutation classes of integers, each identified by its smallest member.

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A014430 Subtract 1 from Pascal's triangle, read by rows.

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A014553 Maximal multiplicative persistence (or length) of any n-digit number.

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A289410 Irregular triangular array T(m,k) with m (row) >= 1 and k (column) >= 1 read by rows: number of m-digit numbers whose digit sum is k.

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A115205 a(n) = binomial(n, 9) + 1.

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A165618 a(n) = binomial(n+8,8) - 1.