Real-rooted atlas

A008957: Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).

A008957 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence from A008957

\(P_{n}(x) = \left(1 + n^{2} x\right)\,P_{n-1}(x) + \left(x^{2} - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21517
311421137
4130147851264
5155627140834112433
6191200211440130131365127913
7114052786149019605311848254611386906

All 15 rows b-files: A008957, A036969 (reversed triangle) ?

Technical details
Node id
n-c23f36ab5968876b · revision 1 · JSON
Definition source
oeis_catalog:A008957
Fingerprint
f6-de3a1cd8e58b208d

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

  • A008957 triangle 50 terms, 9 rows · tabl · nonn · catalog b-file?
  • A036969 reversed triangle 50 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A135920

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   5,     1;
  3: 1,  14,    21,      1;
  4: 1,  30,   147,     85,        1;
  5: 1,  55,   627,   1408,      341,        1;
  6: 1,  91,  2002,  11440,    13013,     1365,        1;
  7: 1, 140,  5278,  61490,   196053,   118482,     5461,       1;
  8: 1, 204, 12138, 251498,  1733303,  3255330,  1071799,   21845,     1;
  9: 1, 285, 25194, 846260, 10787231, 46587905, 53157079, 9668036, 87381, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+n^2*x)*P(n-1,x) - (2*n-1)*x^2*d/dx P(n-1,x) + x^3*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + n^2*x)*P[n - 1, x] - (2*n - 1)*x^2*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + n^2*x)*v[i - 1] - (2*n - 1)*x^2*deriv(v[i - 1], x) + x^3*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A008957 triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 1;
   2: 1,   1;
   3: 1,   5,     1;
   4: 1,  14,    21,      1;
   5: 1,  30,   147,     85,        1;
   6: 1,  55,   627,   1408,      341,        1;
   7: 1,  91,  2002,  11440,    13013,     1365,        1;
   8: 1, 140,  5278,  61490,   196053,   118482,     5461,       1;
   9: 1, 204, 12138, 251498,  1733303,  3255330,  1071799,   21845,     1;
  10: 1, 285, 25194, 846260, 10787231, 46587905, 53157079, 9668036, 87381, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n^2-2*n+1)*x)*P(n-1,x) - (2*n-3)*x^2*d/dx P(n-1,x) + x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (n^2 - 2*n + 1)*x)*P[n - 1, x] - (2*n - 3)*x^2*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n^2 - 2*n + 1)*x)*v[i - 1] - (2*n - 3)*x^2*deriv(v[i - 1], x) + x^3*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A036969 reversed triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 1;
   2: 1,     1;
   3: 1,     5,       1;
   4: 1,    21,      14,        1;
   5: 1,    85,     147,       30,        1;
   6: 1,   341,    1408,      627,       55,        1;
   7: 1,  1365,   13013,    11440,     2002,       91,      1;
   8: 1,  5461,  118482,   196053,    61490,     5278,    140,     1;
   9: 1, 21845, 1071799,  3255330,  1733303,   251498,  12138,   204,   1;
  10: 1, 87381, 9668036, 53157079, 46587905, 10787231, 846260, 25194, 285, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n^2-2*n+1)*x)*P(n-1,x) - (2*n-3)*x^2*d/dx P(n-1,x) + x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (n^2 - 2*n + 1)*x)*P[n - 1, x] - (2*n - 3)*x^2*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n^2 - 2*n + 1)*x)*v[i - 1] - (2*n - 3)*x^2*deriv(v[i - 1], x) + x^3*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Used in 5

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.