A134991: Triangle of Ward numbers T(n,k) read by rows.

OEIS matches ?

• A000311: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A134991: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

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.

Triangle begins:
   1: 1;
   2: 1,    3;
   3: 1,   10,     15;
   4: 1,   25,    105,     105;
   5: 1,   56,    490,    1260,      945;
   6: 1,  119,   1918,    9450,    17325,     10395;
   7: 1,  246,   6825,   56980,   190575,    270270,     135135;
   8: 1,  501,  22935,  302995,  1636635,   4099095,    4729725,    2027025;
   9: 1, 1012,  74316, 1487200, 12122110,  47507460,   94594500,   91891800,   34459425;
  10: 1, 2035, 235092, 6914908, 81431350, 466876410, 1422280860, 2343240900, 1964187225, 654729075;

The row polynomials satisfy P(n,x) = (1+(n+1)*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (n + 1)*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n + 1)*x)*v[i - 1] + (x + x^2)*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, v[i])))};
T(10)

Triangle begins:
   1: 1;
   2: 1,    3;
   3: 1,   10,     15;
   4: 1,   25,    105,     105;
   5: 1,   56,    490,    1260,      945;
   6: 1,  119,   1918,    9450,    17325,     10395;
   7: 1,  246,   6825,   56980,   190575,    270270,     135135;
   8: 1,  501,  22935,  302995,  1636635,   4099095,    4729725,    2027025;
   9: 1, 1012,  74316, 1487200, 12122110,  47507460,   94594500,   91891800,   34459425;
  10: 1, 2035, 235092, 6914908, 81431350, 466876410, 1422280860, 2343240900, 1964187225, 654729075;

The row polynomials satisfy P(n,x) = (1+(n+1)*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (n + 1)*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n + 1)*x)*v[i - 1] + (x + x^2)*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)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-c13d1846cdae7550
• Coefficients of one sign. Every nonzero row, n >= 1: 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-c13d1846cdae7550
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-c13d1846cdae7550
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-c13d1846cdae7550

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

• theorem_check ma_wang_interval e-c13d1846cdae7550 root: theorem check

  Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 2 proof steps

• e-72ea2d9fc8c7fc63 obreschkoff_combination → obreschkoff_combination of Triangle of Ward numbers T(n,k) read by rows.
• e-72ea2d9fc8c7fc63 obreschkoff_combination → obreschkoff_combination of Triangle of Ward numbers T(n,k) read by rows.

Related sequences ?

Same fingerprint (1)

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

• A269939: Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n…

References

1. [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