Real-rooted atlas

A160562: Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).

A160562 ↗ nonnegative coefficients

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

Recurrence from A160562

\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(8\,x\right)\,P_{n-1}'(x) + \left(4\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
2110112
3191351128
418209668411872
517381249705082165137600
6166430631631273988184472861990784
715978711585720514057043176819553053455132333824

All 15 rows b-files: A008958 (reversed triangle), A160562 ?

Technical details
Node id
n-65e47d0258ed4170 · revision 0 · JSON
Definition source
oeis_catalog:A160562
Fingerprint
f6-af987ff96e7c5701

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 ?

  • A008958 reversed triangle 45 terms, 9 rows · tabl · nonn b-file?
  • A160562 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,        1;
  2: 1,       10,          1;
  3: 1,       91,         35,           1;
  4: 1,      820,        966,          84,           1;
  5: 1,     7381,      24970,        5082,         165,          1;
  6: 1,    66430,     631631,      273988,       18447,        286,        1;
  7: 1,   597871,   15857205,    14057043,     1768195,      53053,      455,      1;
  8: 1,  5380840,  397027996,   704652312,   157280838,    8187608,   129948,    680,   1;
  9: 1, 48427561, 9931080740, 34924991284, 13444400190, 1147981406, 30148820, 282948, 969, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 8*x*d/dx P(n-1,x) + 4*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[(1 + x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x] + 4*x^2*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 + x)*v[i - 1] + 8*x*deriv(v[i - 1], x) + 4*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 - 1, v[i])))};
T(9)

polytool c48629374597

A008958 reversed triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,  10,      1;
  3: 1,  35,     91,        1;
  4: 1,  84,    966,      820,          1;
  5: 1, 165,   5082,    24970,       7381,           1;
  6: 1, 286,  18447,   273988,     631631,       66430,           1;
  7: 1, 455,  53053,  1768195,   14057043,    15857205,      597871,          1;
  8: 1, 680, 129948,  8187608,  157280838,   704652312,   397027996,    5380840,        1;
  9: 1, 969, 282948, 30148820, 1147981406, 13444400190, 34924991284, 9931080740, 48427561, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + 8*x*d/dx P(n-1,x) + 4*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[(1 + x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x] + 4*x^2*D[P[n - 1, x], {x, 2}]];
Table[Reverse[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] = (1 + x)*v[i - 1] + 8*x*deriv(v[i - 1], x) + 4*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 = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A160562 triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 1,        1;
  2: 1,       10,          1;
  3: 1,       91,         35,           1;
  4: 1,      820,        966,          84,           1;
  5: 1,     7381,      24970,        5082,         165,          1;
  6: 1,    66430,     631631,      273988,       18447,        286,        1;
  7: 1,   597871,   15857205,    14057043,     1768195,      53053,      455,      1;
  8: 1,  5380840,  397027996,   704652312,   157280838,    8187608,   129948,    680,   1;
  9: 1, 48427561, 9931080740, 34924991284, 13444400190, 1147981406, 30148820, 282948, 969, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 8*x*d/dx P(n-1,x) + 4*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[(1 + x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x] + 4*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] = (1 + x)*v[i - 1] + 8*x*deriv(v[i - 1], x) + 4*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.