A025015 -id:A025015 - cp's OEIS frontend

A077042 Square array read by falling antidiagonals of central polynomial coefficients: largest coefficient in expansion of (1 + x + x^2 + ... + x^(n-1))^k = ((1-x^n)/(1-x))^k, i.e., the coefficient of x^floor(k(n-1)/2) and of x^ceiling(k(n-1)/2); also number of compositions of floor(k*(n+1)/2) into exactly k positive integers each no more than n.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 1, 1, 0, 1, 6, 7, 4, 1, 1, 0, 1, 10, 19, 12, 5, 1, 1, 0, 1, 20, 51, 44, 19, 6, 1, 1, 0, 1, 35, 141, 155, 85, 27, 7, 1, 1, 0, 1, 70, 393, 580, 381, 146, 37, 8, 1, 1, 0, 1, 126, 1107, 2128, 1751, 780, 231, 48, 9, 1, 1, 0, 1, 252, 3139

Offset: 0

Henry Bottomley, Oct 22 2002

From Michel Marcus, Dec 01 2012: (Start)
A pair of numbers written in base n are said to be comparable if all digits of the first number are at least as big as the corresponding digit of the second number, or vice versa. Otherwise, this pair will be defined as uncomparable. A set of pairwise uncomparable integers will be called anti-hierarchic.
T(n,k) is the size of the maximal anti-hierarchic set of integers written with k digits in base n.
For example, for base n=2 and k=4 digits:
- 0 (0000) and 15 (1111) are comparable, while 6 (0110) and 9 (1001) are uncomparable,
- the maximal antihierarchic set is {3 (0011), 5 (0101), 6 (0110), 9 (1001), 10 (1010), 12 (1100)} with 6 elements that are all pairwise uncomparable. (End)

			Rows of square array start:
  1,    0,    0,    0,    0,    0,    0, ...
  1,    1,    1,    1,    1,    1,    1, ...
  1,    1,    2,    3,    6,   10,   20, ...
  1,    1,    3,    7,   19,   51,  141, ...
  1,    1,    4,   12,   44,  155,  580, ...
  1,    1,    5,   19,   85,  381, 1751, ...
  ...
Read by antidiagonals:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 2, 1, 1;
  0, 1, 3, 3, 1, 1;
  0, 1, 6, 7, 4, 1, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Denis Bouyssou, Thierry Marchant, Marc Pirlot, The size of the largest antichains in products of linear orders, arXiv:1903.07569 [math.CO], 2019.
J. W. Sander, On maximal antihierarchic sets of integers, Discrete Mathematics, Volume 113, Issues 1-3, 5 April 1993, Pages 179-189.
Index entries for sequences related to compositions

Rows include A000007, A000012, A001405, A002426, A005190, A005191, A018901, A025012, A025013, A025014, A025015, A201549, A225779, A201550. Columns include A000012, A000012, A001477, A077043, A005900, A077044, A071816, A133458.
Central diagonal is A077045, with A077046 and A077047 either side. Cf. A067059, A270918, A201552.
Cf. A273975.

t[n_, k_] := Max[ CoefficientList[ Series[ ((1-x^n) / (1-x))^k, {x, 0, k*(n-1)}], x]]; t[0, 0] = 1; t[0, ] = 0; Flatten[ Table[ t[n-k, k], {n, 0, 12}, {k, n, 0, -1}]] (* _Jean-François Alcover, Nov 04 2011 *)

T(n,k)=if(n<1 || k<1,k==0,vecmax(Vec(((1-x^n)/(1-x))^k)))

By the central limit theorem, T(n,k) is roughly n^(k-1)*sqrt(6/(Pi*k)).

T(n,k) = Sum{j=0,h/n} (-1)^j*binomial(k,j)*binomial(k-1+h-n*j,k-1) with h=floor(k*(n-1)/2), k>0. - Michel Marcus, Dec 01 2012

A273975 Three-dimensional array written by antidiagonals in k,n: T(k,n,h) with k >= 1, n >= 0, 0 <= h <= n*(k-1) is the coefficient of x^h in the polynomial (1 + x + ... + x^(k-1))^n = ((x^k-1)/(x-1))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 6, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10

Offset: 1

Andrey Zabolotskiy, Nov 10 2016

Equivalently, T(k,n,h) is the number of ordered sets of n nonnegative integers < k with the sum equal to h.
From Juan Pablo Herrera P., Nov 21 2016: (Start)
T(k,n,h) is the number of possible ways of randomly selecting h cards from k-1 sets, each with n different playing cards. It is also the number of lattice paths from (0,0) to (n,h) using steps (1,0), (1,1), (1,2), ..., (1,k-1).
Shallow diagonal sums of each triangle with fixed k give the k-bonacci numbers. (End)
T(k,n,h) is the number of n-dimensional grid points of a k X k X ... X k grid, which are lying in the (n-1)-dimensional hyperplane which is at an L1 distance of h from one of the grid's corners, and normal to the corresponding main diagonal of the grid. - Eitan Y. Levine, Apr 23 2023

			For first few k and for first few n, the rows with h = 0..n*(k-1) are given:
k=1:  1;  1;  1;  1;  1; ...
k=2:  1;  1, 1;  1, 2, 1;  1, 3, 3, 1;  1, 4, 6, 4, 1; ...
k=3:  1;  1, 1, 1;  1, 2, 3, 2, 1;  1, 3, 6, 7, 6, 3, 1; ...
k=4:  1;  1, 1, 1, 1;  1, 2, 3, 4, 3, 2, 1; ...
For example, (1 + x + x^2)^3 = 1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6, hence T(3,3,2) = T(3,3,4) = 6.
From _Eitan Y. Levine_, Apr 23 2023: (Start)
Example for the repeated cumulative sum formula, for (k,n)=(3,3) (each line is the cumulative sum of the previous line, and the first line is the padded, alternating 3rd row from Pascal's triangle):
  1  0  0 -3  0  0  3  0  0 -1
  1  1  1 -2 -2 -2  1  1  1
  1  2  3  1 -1 -3 -2 -1
  1  3  6  7  6  3  1
which is T(3,3,h). (End)

Andrey Zabolotskiy, slices with k+n = 1..20, flattened
Florentin Smarandache, K-Nomial Coefficients, arXiv:math/0612062 [math.GM], 2006 (originally published in French in: F. Smarandache, Généralisations et Généralités, Ed. Nouvelle, 1984, pp. 24-26).

k-nomial arrays for fixed k=1..10: A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.
Arrays for fixed n=0..6: A000012, A000012, A004737, A109439, A277949, A277950, A277951.
Central n-nomial coefficients for n=1..9, i.e., sequences with h=floor(n*(k-1)/2) and fixed n: A000012, A000984 (A001405), A002426, A005721 (A005190), A005191, A063419 (A018901), A025012, (A025013), A025014, A174061 (A025015), A201549, (A225779), A201550. Arrays: A201552, A077042, see also cfs. therein.
Triangle n=k-1: A181567. Triangle n=k: A163181.

a = Table[CoefficientList[Sum[x^(h-1),{h,k}]^n,x],{k,10},{n,0,9}];
Flatten@Table[a[[s-n,n+1]],{s,10},{n,0,s-1}]
(* alternate program *)
row[k_, n_] := Nest[Accumulate,Upsample[Table[((-1)^j)*Binomial[n,j],{j,0,n}],k],n][[;;n*(k-1)+1]] (* Eitan Y. Levine, Apr 23 2023 *)

T(k,n,h) = Sum_{i = 0..floor(h/k)} (-1)^i*binomial(n,i)*binomial(n+h-1-k*i,n-1). [Corrected by Eitan Y. Levine, Apr 23 2023]
From Eitan Y. Levine, Apr 23 2023: (Start)
(T(k,n,h))_{h=0..n*(k-1)} = f(f(...f(g(P))...)), where:
(x_i)_{i=0..m} denotes a tuple (in particular, the LHS contains the values for 0 <= h <= n*(k-1)),
f repeats n times,
f((x_i){i=0..m}) = (Sum{j=0..i} x_j)_{i=0..m} is the cumulative sum function,
g((x_i){i=0..m}) = (x(i/k) if k|i, otherwise 0)_{i=0..m*k} is adding k-1 zeros between adjacent elements,
and P=((-1)^i*binomial(n,i))_{i=0..n} is the n-th row of Pascal's triangle, with alternating signs. (End)
From Eitan Y. Levine, Jul 27 2023: (Start)
Recurrence relations, the first follows from the sequence's defining polynomial as mentioned in the Smarandache link:
T(k,n+1,h) = Sum_{i = 0..s-1} T(k,n,h-i)
T(k+1,n,h) = Sum_{i = 0..n} binomial(n,i)*T(k,n-i,h-i*k) (End)

A174061 The Lucky Tickets Problem.

Original entry on oeis.org

1, 10, 670, 55252, 4816030, 432457640, 39581170420, 3671331273480, 343900019857310, 32458256583753952, 3081918923741896840, 294056694657804068000, 28170312778225750242100

Offset: 0

Geoffrey Critzer, Mar 06 2010, Mar 13 2010

A ticket has a 2n-digit number. (The initial digits are allowed to be zeros.) A ticket is lucky if the sum of the first n digits is equal to the sum of the last n digits. a(n) is the number of lucky tickets. a(n) is also the number of tickets in which the sum of all the digits is 9*n.

a(n) is the number of integers whose digits sum = 9*n in [0, 100^n-1]. The most common value of sums of digits of numbers in [0, 100^n-1] is 9*n. - Miquel Cerda, Jul 02 2017

			The ticket 123051 is lucky because 1 + 2 + 3 = 0 + 5 + 1.
670 is the number of integers in the [0, 100^2-1] range whose digits sum = 18 and 55252 is the number of integers in the [0, 100^3-1] range whose digits sum = 27. - _Miquel Cerda_, Jul 02 2017

S. K. Lando, Lectures on Generating Functions, AMS, 2002, page 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. Dubey, A Simplified Analysis To A Generalized Restricted Partition Problem, Principia: The Princeton Undergraduate Mathematics Journal, Issue 2, 2016.

Table[Total[ CoefficientList[Series[((1 - x^10)/(1 - x))^n, {x, 0, 9*n}], x]^2], {n, 0, 15}]

a(n)=if(n==0, 1, sum(k=0, n - 1, (-1)^k*binomial(2*n, k)*binomial(11*n - 1 - 10*k, 2*n - 1))); \\ Indranil Ghosh, Jul 01 2017

a(n) = Sum_{k=0..n-1} (-1)^k * binomial(2n,k) * binomial(11n-1-10k,2n-1).
a(n) = [x^(9n)] ((1 - x^10)/(1 - x))^(2n).
a(n) = A025015(2*n). - Miquel Cerda, Jul 18 2017

A270918 Largest coefficient of (1+x+...+x^n)^(2*n).

Original entry on oeis.org

1, 2, 19, 580, 38165, 4395456, 786588243, 202384723528, 70886845397481, 32458256583753952, 18832730699014127291, 13507852690353224821652, 11738630472138500287398379, 12155701820213424461220851360, 14790850878997102285050287114419

Offset: 0

Vaclav Kotesovec, Mar 26 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..210

Cf. A077047.

Cf. A002426, A005190, A005191, A018901, A025012, A025013, A025014, A025015.

Table[Max[CoefficientList[Expand[Sum[x^k, {k, 0, n}]^(2n)], x]], {n, 0, 20}]

a(n) = vecmax(Vec((sum(k=0,n,x^k))^(2*n))); \\ Michel Marcus, Apr 01 2016

a(n) ~ exp(2) * sqrt(3/Pi) * n^(2*n - 3/2).

Typo in formula corrected by Vaclav Kotesovec, Dec 10 2021

A197357 Number of n-digits integers for which the sum of the odd-positioned digits equals the sum of the even-positioned digits.

Original entry on oeis.org

1, 10, 55, 670, 4840, 55252, 436975, 4816030, 40051495, 432457640, 3715101654, 39581170420, 347847754670, 3671331273480, 32811494188975, 343900019857310, 3113537578058979, 32458256583753952, 296896918816556380, 3081918923741896840

Offset: 1

Colin Barker, Oct 13 2011

a(n) is the number of n-digit integers such that the sum of the odd-positioned digits is equal to the sum of the even-positioned digits, leading zeros being allowed in the integers.

			The number 28754 is one of the 4840 5-digit numbers because 2+7+4 = 8+5.

Colin Barker, Table of n, a(n) for n = 1..200

This sequence has some numbers in common with both A025015 and A174061. In fact, A174061 consists of the elements a(2n), and the elements a(2n) are all elements of A025015.

a(n) = {nb = 0; for (i=0, 10^n-1, digs = digits(i, 10); while(#digs != n, digs = concat(0, digs)); so = 0; forstep(j=1, n, 2, so += digs[j]); se = 0; forstep(j=2, n, 2, se += digs[j]); if (se == so, nb++);); return (nb);} \\ Michel Marcus, Jun 08 2013

A382531 Number of n-digit base-10 numbers whose digit sum is equal to ceiling(9*n/2).

Original entry on oeis.org

1, 9, 70, 615, 5520, 50412, 468448, 4379055, 41395240, 392406145, 3748943890, 35866068766, 345143007910, 3323483518810, 32150758083580, 311088525668335, 3021445494584902, 29344719005694973, 285904843977651598, 2785022004925340460, 27203012941819689340

Offset: 1

Miquel Cerda, Mar 30 2025

Digit sum ceiling(9*n/2) = A130877(n+1) has highest frequency among all n-digit base-10 numbers.

The count excludes numbers with leading zeros.

			a(2) = 9, the 2-digit numbers with digit sum 9 are: 18, 27, 36, 45, 54, 63, 72, 81, 90.

Alois P. Heinz, Table of n, a(n) for n = 1..1002

Cf. A210736 (analogous for base-2 digits).
Cf. A025015 (maximal coefficient of (1+...+x^9)^n).
Cf. A007953, A071817, A071976, A130877.

b:= proc(n, s, t) option remember; `if`(9*n b(n, ceil(9*n/2), 1):
seq(a(n), n=1..23);  # Alois P. Heinz, Apr 12 2025

a(n) = [x^ceiling(9*n/2)] (f^n - f^(n-1)) with f = (x^10-1)/(x-1). - Alois P. Heinz, Apr 12 2025

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A273975 Three-dimensional array written by antidiagonals in k,n: T(k,n,h) with k >= 1, n >= 0, 0 <= h <= n*(k-1) is the coefficient of x^h in the polynomial (1 + x + ... + x^(k-1))^n = ((x^k-1)/(x-1))^n.

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A174061 The Lucky Tickets Problem.

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A270918 Largest coefficient of (1+x+...+x^n)^(2*n).

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A197357 Number of n-digits integers for which the sum of the odd-positioned digits equals the sum of the even-positioned digits.

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A382531 Number of n-digit base-10 numbers whose digit sum is equal to ceiling(9*n/2).

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