cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A109692 Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 9, 23, 15, 1, 16, 86, 176, 105, 1, 25, 230, 950, 1689, 945, 1, 36, 505, 3480, 12139, 19524, 10395, 1, 49, 973, 10045, 57379, 177331, 264207, 135135, 1, 64, 1708, 24640, 208054, 1038016, 2924172, 4098240, 2027025
Offset: 0

Views

Author

Philippe Deléham, Aug 08 2005

Keywords

Comments

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 2, 3, 4, 5, 6, 7, 8, 9, ...] where DELTA is the operator defined in A084938.
T(n,k), 0 <= k <= n, is the number of elements in the Coxeter group B_n with absolute length k. - Jose Bastidas, Jul 14 2023

Examples

			Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,   3;
  1,  9,  23,   15;
  1, 16,  86,  176,   105;
  1, 25, 230,  950,  1689,   945;
  1, 36, 505, 3480, 12139, 19524, 10395;
  ...

Links

Crossrefs

Cf. A039758 (signed version). A028338 transposed.
Diagonals: A000012, A000290, A024196, A024197, A024198; A001147, A004041, A028339, A028340, A028341.
Row sums: A000165.
Central terms: A293318.
Cf. A161198 (transposed scaled triangle version).

Programs

  • Maple
    nmax:=8; mmax:=nmax: for n from 0 to nmax do a(n, n) := doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, 0):=1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := a(n-1,m) + (2*n-1)*a(n-1,m-1) od; od: seq(seq(a(n, m), m=0..n), n=0..nmax); # Johannes W. Meijer, Jun 08 2009, revised Nov 25 2012

Formula

T(n,m) = T(n-1,m) + (2*n-1)*T(n-1,m-1) with T(n,n) = (2*n-1)!! and T(n,0) = 1. - Johannes W. Meijer, Jun 08 2009

A039762 Triangle of D-analogs of Stirling numbers of first kind.

Original entry on oeis.org

1, 0, 1, 1, -2, 1, -6, 11, -6, 1, 45, -84, 50, -12, 1, -420, 809, -520, 150, -20, 1, 4725, -9390, 6439, -2100, 355, -30, 1, -62370, 127539, -92358, 33019, -6510, 721, -42, 1, 945945, -1984584, 1505524, -578984, 127694, -16856, 1316, -56, 1, -16216200, 34812945, -27491616, 11228300, -2702448, 405174, -38304, 2220, -72, 1
Offset: 0

Views

Author

Ruedi Suter (suter(AT)math.ethz.ch)

Keywords

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
     1;
     0,   1;
     1,  -2,    1;
    -6,  11,   -6,   1;
    45, -84,   50, -12,   1;
  -420, 809, -520, 150, -20, 1;
  ...

Links

Crossrefs

Cf. A039757, A039758, A039763 (transposed triangle).

Programs

  • PARI
    row(n) = if(n==0, [1], Vecrev(prod(i=1, n-1, x-2*i+1)*(x-n+1))); \\ Petros Hadjicostas, Jul 12 2020

Formula

From Petros Hadjicostas, Jul 11 2020: (Start)
T(n,k) = [x^k] (x - (n - 1)) * Product_{k=1..n-1} (x - (2*k - 1)) for n >= 1 with T(0,0) = 1. (Empty products equal 1.)
Let R(n,k) = A039757(n,k) = A039758(n,n-k). Then, for n >= 1:
T(n,0) = -(n - 1)*R(n-1,0);
T(n,k) = R(n-1,k-1) - (n - 1)*R(n-1,k) for k = 1..n-1;
T(n,n) = R(n-1, n-1) = 1.
As a result, for n >= 2, T(n,0) = (-1)^n*(n-1)*(2*n-3)!!. (End)

Extensions

More terms from Petros Hadjicostas, Jul 12 2020

A349226 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n} (x - k^k) expanded in decreasing powers of x, with row 0 = {1}.

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -6, 9, -4, 1, -33, 171, -247, 108, 1, -289, 8619, -44023, 63340, -27648, 1, -3413, 911744, -26978398, 137635215, -197965148, 86400000, 1, -50070, 160195328, -42565306462, 1258841772303, -6421706556188, 9236348345088, -4031078400000
Offset: 0

Views

Author

Thomas Scheuerle, Jul 07 2022

Keywords

Comments

Let M be an n X n matrix filled by binomial(i*j, i) with rows and columns j = 1..n, k = 1..n; then its determinant equals unsigned T(n, n). Can we find a general formula for T(n+m, n) based on determinants of matrices and binomials?

Examples

			The triangle begins:
  1;
  1,    -1;
  1,    -2,      1;
  1,    -6,      9,        -4;
  1,   -33,    171,      -247,       108;
  1,  -289,   8619,    -44023,     63340,     -27648;
  1, -3413, 911744, -26978398, 137635215, -197965148, 86400000;
  ...
Row 4: x^4-33*x^3+171*x^2-247*x+108 = (x-1)*(x-1^1)*(x-2^2)*(x-3^3).

Crossrefs

Cf. A002109, A062970.
Cf. A008276 (The Stirling numbers of the first kind in reverse order).
Cf. A039758 (Coefficients for polynomials with roots in odd numbers).
Cf. A355540 (Coefficients for polynomials with roots in factorials).

Programs

  • PARI
    T(n, k) = polcoeff(prod(m=0, n-1, (x-m^m)), n-k);

Formula

T(n, 0) = 1.
T(n, 1) = -A062970(n).
T(n, 2) = Sum_{m=0..n-1} A062970(m)*m^m.
T(n, k) = Sum_{m=0..n-1} -T(m, k-1)*m^m.
T(n, n) = (-1)^n*A002109(n).

A355540 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n} (x - k!) expanded in decreasing powers of x, with row 0 = {1}.

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -10, 29, -32, 12, 1, -34, 269, -728, 780, -288, 1, -154, 4349, -33008, 88140, -93888, 34560, 1, -874, 115229, -3164288, 23853900, -63554688, 67633920, -24883200, 1, -5914, 4520189, -583918448, 15971865420, -120287210688, 320383261440, -340899840000, 125411328000
Offset: 0

Views

Author

Thomas Scheuerle, Jul 06 2022

Keywords

Comments

Essentially the same as A136457 with rows in reversed order.
Let M be an n X n matrix filled by Bell numbers A000110(j+k-2) with rows and columns j = 1..n, k = 1..n; then its determinant equals unsigned T(n, n). If we use A000110(j+k), the determinant will equal unsigned T(n+1, n). Can we find a general formula for T(n+m, n) based on determinants of matrices and Bell numbers?

Examples

			The triangle begins:
  1;
  1,   -1;
  1,   -2,      1;
  1,   -4,      5,       -2;
  1,  -10,     29,      -32,       12;
  1,  -34,    269,     -728,      780,      -288;
  1, -154,   4349,   -33008,    88140,    -93888,    34560;
  1, -874, 115229, -3164288, 23853900, -63554688, 67633920, -24883200;
  ...
Row 4: x^4 - 10*x^3 + 29*x^2 - 32*x + 12 = (x-0!)*(x-1!)*(x-2!)*(x-3!).
Illustration of T(1 to 5,1) as tree structure:
.
. o        o         o            o                         o
.          o         o            o                         o
.                   o o          o o                       o o
.                              ooo ooo                   ooo ooo
.                                             oooo oooo oooo oooo oooo oooo
. 1 +1 =   2 +2 =    4 +2*3 =     10 +6*4 =                 34
.
Illustration of T(2 to 4,2) as tree structure:
.
. o         o              -----o-----
.        o     o          o           o
.        o     o       ---o---     ---o---
.                     o   o   o   o   o   o
.                     o   o   o   o   o   o
.                    o o o o o o o o o o o o
. 1 +2*2 =  5 +6*4 =            29
.
Illustration of T(3 to 4,3) as tree structure:
.            ------------
. oo     ---o---      ---o---
.       o   o   o    o   o   o
.      o o o o o o  o o o o o o
.      o o o o o o  o o o o o o
.  2  +6*5 =      32

Crossrefs

Cf. A000110, A000178, A000522, A003422, A136457, A203227, A217757.
Cf. A008276 (The Stirling numbers of the first kind in reverse order).
Cf. A039758 (Coefficients for polynomials with roots in odd numbers).
Cf. A349226 (Coefficients for polynomials with roots in x^x).

Programs

  • PARI
    T(n, k) = polcoeff(prod(m=0, n-1, (x-m!)), n-k);

Formula

T(n, 0) = 1.
T(n, 1) = -A003422(n).
T(n, 2) = Sum_{m=0..n-1} !m*m!.
T(n, k) = Sum_{m=0..n-1} -T(m, k-1)*m!.
T(n, n) = (-1)^n*A000178(n).
T(n, n-1) = -(-1)^n*A203227(n), for n > 0.
T(n+1, n) = (-1)^n*A000178(n)*A000522(n).
Sum_{m=0..k} T(n, k) = 0, for n > 0.
Sum_{m=0..k} abs(T(n, k)) = A217757(n+1).
Showing 1-4 of 4 results.

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