Real-rooted atlas

A191935: Triangle read by rows of Legendre-Stirling numbers of the second kind.

A191935 ↗ nonnegative coefficients

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

Recurrence from A191935

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
218413
312052881
414029232016669
5170109238241936326955
6111231922566447824116486488505
711687896121424561104585536699521281346209

All 15 rows b-files: A071951 (reversed triangle), A191935 ?

Technical details
Node id
n-1dda68ade0f78df3 · revision 0 · JSON
Definition source
oeis_catalog:A191935
Fingerprint
f6-57cb71c89ecc0cb7

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 ?

  • A071951 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
  • A191935 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A135921

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,   2;
  2: 1,   8,     4;
  3: 1,  20,    52,       8;
  4: 1,  40,   292,     320,       16;
  5: 1,  70,  1092,    3824,     1936,        32;
  6: 1, 112,  3192,   25664,    47824,     11648,        64;
  7: 1, 168,  7896,  121424,   561104,    585536,     69952,      128;
  8: 1, 240, 17304,  453056,  4203824,  11807616,   7096384,   419840,     256;
  9: 1, 330, 34584, 1422080, 23232176, 137922336, 243248704, 85576448, 2519296, 512;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n^2+n)*x)*P(n-1,x) - 2*n*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 + n)*x)*P[n - 1, x] - 2*n*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 + n)*x)*v[i - 1] - 2*n*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

A071951 reversed triangle, offset 1, from P0
%e Example
Triangle begins:
   1:   1;
   2:   2,       1;
   3:   4,       8,        1;
   4:   8,      52,       20,         1;
   5:  16,     320,      292,        40,         1;
   6:  32,    1936,     3824,      1092,        70,        1;
   7:  64,   11648,    47824,     25664,      3192,      112,       1;
   8: 128,   69952,   585536,    561104,    121424,     7896,     168,     1;
   9: 256,  419840,  7096384,  11807616,   4203824,   453056,   17304,   240,   1;
  10: 512, 2519296, 85576448, 243248704, 137922336, 23232176, 1422080, 34584, 330, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n^2-n)*x)*P(n-1,x) - (2*n-2)*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 - n)*x)*P[n - 1, x] - (2*n - 2)*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 - n)*x)*v[i - 1] - (2*n - 2)*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

A191935 triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 1;
   2: 1,   2;
   3: 1,   8,     4;
   4: 1,  20,    52,       8;
   5: 1,  40,   292,     320,       16;
   6: 1,  70,  1092,    3824,     1936,        32;
   7: 1, 112,  3192,   25664,    47824,     11648,        64;
   8: 1, 168,  7896,  121424,   561104,    585536,     69952,      128;
   9: 1, 240, 17304,  453056,  4203824,  11807616,   7096384,   419840,     256;
  10: 1, 330, 34584, 1422080, 23232176, 137922336, 243248704, 85576448, 2519296, 512;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n^2-n)*x)*P(n-1,x) - (2*n-2)*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 - n)*x)*P[n - 1, x] - (2*n - 2)*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 - n)*x)*v[i - 1] - (2*n - 2)*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

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.