Real-rooted atlas

A130534 → basis A008277

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
266113
3243612173
4120240120201501
5720180012003003014051
650401512012600420063042137633
74032014112014112058800117601176561394353
8362880145152016934408467202116802822420167214596553
93628800163296002177280012700800381024063504060480324090158941091
10399168001995840002993760001995840006985440013970880166320011880049501101824073141
1147900160026345088004390848000329313600013172544003073593604390848039204002178007260132112470162233
12622702080037362124800684972288005708102400025686460800684972288011416204801223164808494200377520102961561202976401213
138717829120056665889280011333177856001038874636800519437318400155831195520296821324803710266560309188880171771606246241419618213535017524403
14130767436800091537205760001983306124800019833061248000109081836864003636061228800779155977600111307996800108216108007214407203279276099372019110210165573803186921

Generated b-files: A089231 (reversed triangle), A105278 ?

Computed in your browser ?

Technical details
Node id
n-011d67b73ad91983 · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-8fea465cea2922a0

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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       2,        1;
  2:       6,        6,        1;
  3:      24,       36,       12,        1;
  4:     120,      240,      120,       20,       1;
  5:     720,     1800,     1200,      300,      30,      1;
  6:    5040,    15120,    12600,     4200,     630,     42,     1;
  7:   40320,   141120,   141120,    58800,   11760,   1176,    56,    1;
  8:  362880,  1451520,  1693440,   846720,  211680,  28224,  2016,   72,  1;
  9: 3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+1+x)*P(n-1,x) + x*d/dx 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 + 1 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
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] = (n + 1 + x)*v[i - 1] + x*deriv(v[i - 1], 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)

Generated with polytool c48629374597.

A089231: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  6,    6;
   4: 1, 12,   36,    24;
   5: 1, 20,  120,   240,    120;
   6: 1, 30,  300,  1200,   1800,     720;
   7: 1, 42,  630,  4200,  12600,   15120,     5040;
   8: 1, 56, 1176, 11760,  58800,  141120,   141120,    40320;
   9: 1, 72, 2016, 28224, 211680,  846720,  1693440,  1451520,   362880;
  10: 1, 90, 3240, 60480, 635040, 3810240, 12700800, 21772800, 16329600, 3628800;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n+x)*P(n-1,x) + x*d/dx 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 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
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] = (n + x)*v[i - 1] + x*deriv(v[i - 1], 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)

Generated with polytool c48629374597.

A105278: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:       1;
   2:       2,        1;
   3:       6,        6,        1;
   4:      24,       36,       12,        1;
   5:     120,      240,      120,       20,       1;
   6:     720,     1800,     1200,      300,      30,      1;
   7:    5040,    15120,    12600,     4200,     630,     42,     1;
   8:   40320,   141120,   141120,    58800,   11760,   1176,    56,    1;
   9:  362880,  1451520,  1693440,   846720,  211680,  28224,  2016,   72,  1;
  10: 3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+x)*P(n-1,x) + x*d/dx 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 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
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] = (n + x)*v[i - 1] + x*deriv(v[i - 1], 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-31e8b29857f291e5
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-31e8b29857f291e5
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-31e8b29857f291e5

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741