Real-rooted atlas

A088996: Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.

A088996 nonnegative coefficients only rational roots

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A088996
\(P_{n}(x) = \left(-1 + n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20123
3027615
406294624105
5024146329326120945
6012087425213604255672010395
707206084212443927140564222125040135135

Show all 15 stored rows

Generated b-files: A001147 (row sums), A059364 (reversed triangle), A088996 ?

Computed in your browser ?

Other recurrences

X ↦ x/(x+1) substitution · proved equal for all n (proof step)

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 - x + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-849de7332df28657 · revision 2 · JSON
Definition source
oeis_catalog:A088996
derived:mobius_x_over_x_plus_one (equal by e-046150b77085a263)
Fingerprint
f6-774e45ab36bcae1f

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): yes

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: 0,     1;
  2: 0,     1,      2;
  3: 0,     2,      7,       6;
  4: 0,     6,     29,      46,      24;
  5: 0,    24,    146,     329,     326,     120;
  6: 0,   120,    874,    2521,    3604,    2556,     720;
  7: 0,   720,   6084,   21244,   39271,   40564,   22212,    5040;
  8: 0,  5040,  48348,  197380,  444849,  598116,  479996,  212976,   40320;
  9: 0, 40320, 432144, 2014172, 5335212, 8788569, 9223012, 6023772, 2239344, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (n-1+n*x)*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 + n*x)*P[n - 1, 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 + n*x)*v[i - 1]); 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.

A059364: reversed triangle, offset 1, from the row P1, low zeros dropped
%e Example
Triangle begins:
   1:       1;
   2:       2,        1;
   3:       6,        7,        2;
   4:      24,       46,       29,         6;
   5:     120,      326,      329,       146,        24;
   6:     720,     2556,     3604,      2521,       874,       120;
   7:    5040,    22212,    40564,     39271,     21244,      6084,      720;
   8:   40320,   212976,   479996,    598116,    444849,    197380,    48348,     5040;
   9:  362880,  2239344,  6023772,   9223012,   8788569,   5335212,  2014172,   432144,   40320;
  10: 3628800, 25659360, 80391816, 146444068, 170892798, 132449241, 68158628, 22448988, 4292496, 362880;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = (n-1+n*x)*P(n-1,x) for n >= 2, with P(1,x) = x.
%t Mathematica
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[(n - 1 + n*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = (n - 1 + n*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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.

A088996: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 0,     1;
  2: 0,     1,      2;
  3: 0,     2,      7,       6;
  4: 0,     6,     29,      46,      24;
  5: 0,    24,    146,     329,     326,     120;
  6: 0,   120,    874,    2521,    3604,    2556,     720;
  7: 0,   720,   6084,   21244,   39271,   40564,   22212,    5040;
  8: 0,  5040,  48348,  197380,  444849,  598116,  479996,  212976,   40320;
  9: 0, 40320, 432144, 2014172, 5335212, 8788569, 9223012, 6023772, 2239344, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (n-1+n*x)*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 + n*x)*P[n - 1, x]];
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 - 1 + n*x)*v[i - 1]); 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)

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-4c0875826685c298
  • 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-4c0875826685c298, e-e5c0b6618046e013
  • 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-4c0875826685c298, e-e5c0b6618046e013
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-e5c0b6618046e013

Proof chains

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

Used in 5 proof steps

Same fingerprint (2)

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