A071919 -id:A071919 - cp's OEIS frontend

A129186 Right shift operator generating 1's in shifted spaces.

1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gary W. Adamson, Apr 01 2007

Let A129186 = M, then M*V, V a vector; shifts V to the right, appending 1's to the shifted spaces. Example: M*V, V = [1,2,3,...] = [1,1,2,3,...].

Triangle T(n,k), read by rows, given by (1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 08 2011

			First few rows of the triangle are:
1;
1, 0;
0, 1, 0;
0, 0, 1, 0;
0, 0, 0, 1, 0;
...

Robert Price, Table of n, a(n) for n = 0..5049

Generalized Eulerian triangles: this sequence (m=0), A173018 (m=1), A292604 (m=2).

Cf. A000012 (row sums), A071919, A129184, A129185.

gf := 1 + z/(1 - x*z): ser := series(gf, z, 16): c := k -> coeff(ser, z, k):
seq(seq(coeff(c(n), x, k), k=0..n), n=0..14); # Peter Luschny, Jul 07 2019

Join[{1},Flatten[Table[PadLeft[{1,0},n,0],{n,2,20}]]] (* Harvey P. Dale, Aug 26 2019 *)

Infinite lower triangular matrix with (1,0,0,...) in the main diagonal and (1,1,1...) in the subdiagonal.

G.f.: (1-(y-1)*x)/(1-y*x). - Philippe Deléham, Dec 08 2011

A132200 Numbers in (4,4)-Pascal triangle .

Original entry on oeis.org

1, 4, 4, 4, 8, 4, 4, 12, 12, 4, 4, 16, 24, 16, 4, 4, 20, 40, 40, 20, 4, 4, 24, 60, 80, 60, 24, 4, 4, 28, 84, 140, 140, 84, 28, 4, 4, 32, 112, 224, 280, 224, 112, 32, 4, 4, 36, 144, 336, 504, 504, 336, 144, 36, 4, 4, 40, 180, 480, 840, 1008, 840, 480, 180, 40, 4

Offset: 0

Philippe Deléham, Nov 19 2007

This triangle belongs to the family of (x,y)-Pascal triangles ; other triangles arise by choosing different values for (x,y): (1,1) -> A007318 ; (1,0) -> A071919 ; (3,2) -> A029618 ; (2,2) -> A134058 ; (-1,1) -> A112467 ; (0,1) -> A097805 ; (5,5) -> A135089 ; etc..

			Triangle begins:
  1;
  4,  4;
  4,  8,  4;
  4, 12, 12,  4;
  4, 16, 24, 16,  4;
  4, 20, 40, 40, 20, 4;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000079, A007318, A134058, A134059, A135089.

[1] cat [4*Binomial(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021

Table[4*Binomial[n,k] -3*Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 03 2021 *)

def A132200(n,k): return 4*binomial(n,k) - 3*bool(n==0)
flatten([[A132200(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

T(n,k) = 4*binomial(n,k), n>0 ; T(0,0)=1.

Sum_{k=0..n} T(n,k) = 2^(n+2) - 3*[n=0]. - G. C. Greubel, May 03 2021

A183190 Triangle T(n,k), read by rows, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 4, 1, 0, 8, 12, 6, 1, 0, 16, 32, 24, 8, 1, 0, 32, 80, 80, 40, 10, 1, 0, 64, 192, 240, 160, 60, 12, 1, 0, 128, 448, 672, 560, 280, 84, 14, 1, 0, 256, 1024, 1792, 1792, 1120, 448, 112, 16, 1, 0, 512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1, 0

Offset: 0

Philippe Deléham, Dec 14 2011

A071919*A007318 as infinite lower triangular matrices.
A129186*A038207 as infinite lower triangular matrices.
From Paul Curtz, Nov 12 2019: (Start)
If a new main diagonal of 0's is added to the triangle, then for this variant the following propositions hold:
   The first column is A166444.
   The second column is A139756.
   The antidiagonal sums are A000129 (Pell numbers).
   The row sums are (-1)^n*A141413.
   The signed row sums are 0 followed by 1's, autosequence companion to A054977.
(End)

			Triangle begins:
   1;
   1,  0;
   2,  1,  0;
   4,  4,  1,  0;
   8, 12,  6,  1,  0;
  16, 32, 24,  8,  1, 0;
  32, 80, 80, 40, 10, 1, 0;
  ...

Essentially the same as A038207, A062715, A065109.
Cf. A001787, A001788, A139756, A000129 (antidiagonals sums).
Cf. A141413, A166444, A054977.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n<2, 1-k, 2*T(n-1, k) +T(n-1, k-1)))
    end:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Nov 08 2019

T[n_, k_] /; 0 <= k <= n := T[n, k] = 2 T[n-1, k] + T[n-1, k-1];
T[0, 0] = T[1, 0] = 1; T[1, 1] = 0; T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 08 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0)=T(1,0)=1 and T(1,1)=0 .
G.f.: (1-(1+y)*x)/(1-(2+y)*x).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A019590(n+1), A000012(n), A011782(n), A133494(n) for x = -2, -1, 0, 1 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000007(n), A133494(n), A020699(n) for x = 0, 1, 2 respectively.
T(2n,n) = A069720(n).

A381425 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of (1 + x/(1-x)^k)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 5, 1, 0, 1, 4, 12, 10, 1, 0, 1, 5, 22, 37, 18, 1, 0, 1, 6, 35, 92, 102, 30, 1, 0, 1, 7, 51, 185, 345, 258, 47, 1, 0, 1, 8, 70, 326, 880, 1188, 606, 70, 1, 0, 1, 9, 92, 525, 1881, 3851, 3796, 1335, 100, 1, 0, 1, 10, 117, 792, 3563, 10002, 15655, 11364, 2781, 138, 1, 0

Offset: 0

Seiichi Manyama, Apr 07 2025

			Square array begins:
  1, 1,  1,   1,    1,     1,     1, ...
  0, 1,  2,   3,    4,     5,     6, ...
  0, 1,  5,  12,   22,    35,    51, ...
  0, 1, 10,  37,   92,   185,   326, ...
  0, 1, 18, 102,  345,   880,  1881, ...
  0, 1, 30, 258, 1188,  3851, 10002, ...
  0, 1, 47, 606, 3796, 15655, 49468, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A000007, A000012, A177787.
Rows n=0..3 give A000012, A001477, A000326, A096000(k-1).
Main diagonal gives A382859.
Cf. A071919, A362125.

a(n, k) = sum(j=0, k, binomial(k, j)*binomial(n+(k-1)*j-1, n-j));

A(n,k) = Sum_{j=0..k} binomial(k,j) * binomial(n+(k-1)*j-1,n-j).

A351580 a(n) is the number of multisets of size n-1 consisting of permutations of n elements.

Original entry on oeis.org

1, 2, 21, 2600, 9078630, 1634935320144, 22831938997720867560, 34390564970975286088924022400, 7457911916650283082000186530740981347120, 300682790088737748950725540713718365319268411170195200, 2830053444386286847574443631356044745870287426798365860653876609636480

Offset: 1

Dan Eilers, Feb 13 2022

nonn

a(n) is the number of reduced men's ranking tables in the stable marriage problem of order n. In the SMP (as noted in A351409), relabeling men or women has no effect on the number of stable matchings. So the women can be relabeled to normalize the order of man #1's rankings (with woman #1 as his first choice and woman n as his last choice), and then the men except man #1 can be relabeled to normalize the lexicographic order of those men's rankings. Since man #1's rankings end up fixed in natural order, they do not contribute to the number of possibilities, leaving n! multichoose (n-1) ways to arrange the rankings of the other n-1 men.
The number of unreduced men's ranking tables is given by A036740. Relabeling just the women reduces this to A134366. Alternately, relabeling just the men reduces A036740 to A344690. Relabeling both men and women reduces the men's relabeling reduction, A344690, by a factor of (n!+n-1)/n to a(n).
It might be tempting to try to reduce A344690 by a factor of n!, but that doesn't work because not all of man #1's rankings are equally likely after relabeling all the men to give man #1 the lexicographically least rankings.
There is room for further relabeling reduction from a(n), given by A263921. The reduction from a(n) to A263921 is analogous to the reduction from reduced latin squares, A000315, to A123234.
Each of the a(n) reduced men's ranking tables can be combined with the A036740 possible unreduced women's ranking tables to form complete instances, but these instances have more possibilities than A351409. For example, a(3)*A036740(3)=21*216=4536 > A351409(3)=3888. However, fewer possibilities result from using A263921 in place of a(n), although the men's ranking tables of A263921 may not be as straightforward to generate. With A263921(3)=10, 10*216=2160 < 3888.

			Starting with the following men's ranking table of order 3, where row k represents man k's rankings, the 1 in the 2nd position of row 3 means that man #3 ranks woman #2 as his 1st choice.
  213
  321
  213
Step 1: reorder columns so row 1 is in natural order:
  123
  231
  123
Step 2: reorder rows 2 to n so rows are in lexical order:
  123
  123
  231
a(3)=21 because there are 1+2+3+4+5+6 = 21 possibilities for the last two rows in lexical order, with 3!=6 possible permutations for each row.
The 21 tables for a(3) are the following:
  123   123   123   123   123   123   123
  123   123   123   123   123   123   132
  123   132   213   231   312   321   132
.
  123   123   123   123   123   123   123
  132   132   132   132   213   213   213
  213   231   312   321   213   231   312
.
  123   123   123   123   123   123   123
  213   231   231   231   312   312   321
  321   231   312   321   312   321   321

Alois P. Heinz, Table of n, a(n) for n = 1..31
Wikipedia, Counting multisets

Cf. A036740, A071919, A134366, A344690, A263921, A351409, A000315, A185141, A030495.

Table[Binomial[n!+n-2,n-1],{n,15}] (* Harvey P. Dale, Jun 02 2023 *)

a(n) = binomial(n! + n - 2, n - 1) \\ Andrew Howroyd, Feb 13 2022

a(n) = binomial(n! + n - 2, n - 1).
a(n) = n*A344690(n)/A030495(n-1).
a(n) = A344690*n/(n! + n - 1).
a(n) = A071919(n-1,n!). - Alois P. Heinz, Feb 16 2022

Erroneous Mathematica program deleted by N. J. A. Sloane, Jun 02 2023

A378320 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(2n-2r+k,r) * binomial(r,n-r)/(2n-2r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 3, 0, 1, 5, 10, 13, 11, 6, 0, 1, 6, 15, 24, 27, 22, 11, 0, 1, 7, 21, 40, 55, 57, 44, 22, 0, 1, 8, 28, 62, 100, 124, 121, 90, 44, 0, 1, 9, 36, 91, 168, 241, 278, 258, 187, 90, 0, 1, 10, 45, 128, 266, 432, 570, 620, 555, 392, 187, 0

Offset: 0

Seiichi Manyama, Nov 23 2024

			Square array begins:
  1,  1,  1,   1,   1,   1,    1, ...
  0,  1,  2,   3,   4,   5,    6, ...
  0,  1,  3,   6,  10,  15,   21, ...
  0,  2,  6,  13,  24,  40,   62, ...
  0,  3, 11,  27,  55, 100,  168, ...
  0,  6, 22,  57, 124, 241,  432, ...
  0, 11, 44, 121, 278, 570, 1077, ...

Columns k=0..1 give A000007, A007477.

Cf. A071919, A378321.

T(n, k, t=0, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x^2 * A_k(x)^(2/k) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A007477.
B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x^2 * B(x)^(k+1). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-2,k+1) for n > 1.

A378321 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3n-3r+k,r) * binomial(r,n-r)/(3n-3r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 3, 0, 1, 4, 6, 8, 6, 0, 1, 5, 10, 16, 19, 16, 0, 1, 6, 15, 28, 42, 50, 42, 0, 1, 7, 21, 45, 79, 114, 137, 114, 0, 1, 8, 28, 68, 135, 224, 322, 380, 322, 0, 1, 9, 36, 98, 216, 401, 652, 918, 1088, 918, 0, 1, 10, 45, 136, 329, 672, 1205, 1912, 2673, 3152, 2673, 0

Offset: 0

Seiichi Manyama, Nov 23 2024

			Square array begins:
  1,  1,   1,   1,   1,    1,    1, ...
  0,  1,   2,   3,   4,    5,    6, ...
  0,  1,   3,   6,  10,   15,   21, ...
  0,  3,   8,  16,  28,   45,   68, ...
  0,  6,  19,  42,  79,  135,  216, ...
  0, 16,  50, 114, 224,  401,  672, ...
  0, 42, 137, 322, 652, 1205, 2088, ...

Columns k=0..1 give A000007, A019497.

Cf. A071919, A378320.

T(n, k, t=0, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x^2 * A_k(x)^(3/k) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A019497.
B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x^2 * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-2,k+2) for n > 1.

A080234 Triangle whose rows are the differences of consecutive pairs of row elements of A080232.

Original entry on oeis.org

1, 1, -2, 1, -1, -1, 1, 0, -2, 0, 1, 1, -2, -2, 1, 1, 2, -1, -4, -1, 2, 1, 3, 1, -5, -5, 1, 3, 1, 4, 4, -4, -10, -4, 4, 4, 1, 5, 8, 0, -14, -14, 0, 8, 5, 1, 6, 13, 8, -14, -28, -14, 8, 13, 6, 1, 7, 19, 21, -6, -42, -42

Offset: 0

Paul Barry, Feb 09 2003

Row sums are -1. Second row pairwise difference of A071919.

			Rows are {1}, {1,-2}, {1,-1,-1}, {1,0,-2,0}, {1,1,-2,-2,1}, ...

Cf. A080232, A071919.

cp's OEIS Frontend

A129186 Right shift operator generating 1's in shifted spaces.

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A132200 Numbers in (4,4)-Pascal triangle .

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A183190 Triangle T(n,k), read by rows, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A381425 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of (1 + x/(1-x)^k)^k.

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PARI

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A351580 a(n) is the number of multisets of size n-1 consisting of permutations of n elements.

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A378320 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(2*n-2*r+k,r) * binomial(r,n-r)/(2*n-2*r+k) for k > 0.

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A378321 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3*n-3*r+k,r) * binomial(r,n-r)/(3*n-3*r+k) for k > 0.

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A080234 Triangle whose rows are the differences of consecutive pairs of row elements of A080232.

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A378320 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(2n-2r+k,r) * binomial(r,n-r)/(2n-2r+k) for k > 0.

A378321 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3n-3r+k,r) * binomial(r,n-r)/(3n-3r+k) for k > 0.