Real-rooted atlas

A008306: Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).

A008306 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence from A008306

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

Rows ?

nx0x1x2x3Pn(1)
011
122
2639
3242044
412013015265
57209242101854
650407308238010514833
74032064224264322520133496

All 15 rows b-files: A008306 ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

\(P_{n}(x) = \left(x + n x\right)\,P_{n-2}(x) + \left(1 + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)
Technical details
Node id
n-f13c6b611f4ce3bb · revision 1 · JSON
Definition source
oeis_catalog:A008306
derived:reversal (equal by e-2bf4ffda16430b0b)
Fingerprint
f6-adb63b75c11d63d7

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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 ?

  • A008306 triangle 36 terms, 11 rows · tabf · nonn · catalog b-file?

Row sums ? A000166, A260115, A260216, A257953

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       2;
  2:       6,       3;
  3:      24,      20;
  4:     120,     130,      15;
  5:     720,     924,     210;
  6:    5040,    7308,    2380,    105;
  7:   40320,   64224,   26432,   2520;
  8:  362880,  623376,  303660,  44100,   945;
  9: 3628800, 6636960, 3678840, 705320, 34650;
%F Formula
The row polynomials satisfy P(n,x) = (n+1)*P(n-1,x) + (n+1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(n + 1)*P[n - 1, x] + (n + 1)*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (n + 1)*v[i - 1] + (n + 1)*x*v[i - 2]); 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

A008306 triangle, offset 2, from P0
%e Example
Triangle begins:
   2:       1;
   3:       2;
   4:       6,       3;
   5:      24,      20;
   6:     120,     130,      15;
   7:     720,     924,     210;
   8:    5040,    7308,    2380,    105;
   9:   40320,   64224,   26432,   2520;
  10:  362880,  623376,  303660,  44100,   945;
  11: 3628800, 6636960, 3678840, 705320, 34650;
%F Formula
The row polynomials satisfy P(n,x) = (n-1)*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 4, with P(2,x) = 1, P(3,x) = 2.
%t Mathematica
P[2, x_] := 1;
P[3, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(n - 1)*P[n - 1, x] + (n - 1)*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 2, 11}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 1)); v[1] = 1; v[2] = 2; for(n = 4, N, my(i = n - 1); v[i] = (n - 1)*v[i - 1] + (n - 1)*x*v[i - 2]); 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(11)

polytool c48629374597

Properties

Proof chains ? JSON

Used in 10

2 more

Same fingerprint ?

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003