Real-rooted atlas

A390433: Triangle read by rows, Lah numbers with level 2.

A390433 ↗ nonnegative coefficients

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

Recurrence from A390433

\(P_{n}(x) = \left(4 + x - 4\,n + n^{2}\right)\,P_{n-1}(x) + \left(-3\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2011
30415
403620157
505765365611169
6014400198723280120137673
705184009881282297921300022011749541
802540160063758592196012801529792396203641110331249

All 15 rows b-files: A390433, A390667 (row sums) ?

Technical details
Node id
n-313c8ccf28f71545 · revision 0 · JSON
Definition source
oeis_catalog:A390433
Fingerprint
f6-17ce5c7cd2025ccd

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 -1)
  • 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 ?

  • A390433 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A390667

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0,            1;
   3: 0,            4,            1;
   4: 0,           36,           20,            1;
   5: 0,          576,          536,           56,           1;
   6: 0,        14400,        19872,         3280,         120,          1;
   7: 0,       518400,       988128,       229792,       13000,        220,        1;
   8: 0,     25401600,     63758592,     19601280,     1529792,      39620,      364,      1;
   9: 0,   1625702400,   5189847552,   2023886592,   204706112,    7235072,   101136,    560,   1;
  10: 0, 131681894400, 520610457600, 250080125184, 31501566720, 1427433280, 27057728, 227136, 816, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n^2-4*n+4+x)*P(n-1,x) + (2*n-3)*x*d/dx P(n-1,x) + x^2*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[(n^2 - 4*n + 4 + x)*P[n - 1, x] + (2*n - 3)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {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] = (n^2 - 4*n + 4 + x)*v[i - 1] + (2*n - 3)*x*deriv(v[i - 1], x) + x^2*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, v[i])))};
T(10)

polytool c48629374597

A390433 triangle, offset 0, from P1
%e Example
Triangle begins:
  0: 1;
  1: 0,            1;
  2: 0,            4,            1;
  3: 0,           36,           20,            1;
  4: 0,          576,          536,           56,           1;
  5: 0,        14400,        19872,         3280,         120,          1;
  6: 0,       518400,       988128,       229792,       13000,        220,        1;
  7: 0,     25401600,     63758592,     19601280,     1529792,      39620,      364,      1;
  8: 0,   1625702400,   5189847552,   2023886592,   204706112,    7235072,   101136,    560,   1;
  9: 0, 131681894400, 520610457600, 250080125184, 31501566720, 1427433280, 27057728, 227136, 816, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n^2-2*n+1+x)*P(n-1,x) + (2*n-1)*x*d/dx P(n-1,x) + x^2*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[(n^2 - 2*n + 1 + x)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (n^2 - 2*n + 1 + x)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

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.