A112486: Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.

OEIS matches ?

• A032034: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A112487: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A112486: triangle, 41 terms agree, 8 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

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:
  0:      1;
  1:      1,       1;
  2:      2,       5,        3;
  3:      6,      26,       35,        15;
  4:     24,     154,      340,       315,       105;
  5:    120,    1044,     3304,      4900,      3465,       945;
  6:    720,    8028,    33740,     70532,     78750,     45045,     10395;
  7:   5040,   69264,   367884,   1008980,   1571570,   1406790,    675675,    135135;
  8:  40320,  663696,  4302216,  14777620,  29957620,  37147110,  27747720,  11486475,   2027025;
  9: 362880, 6999840, 53961336, 224655816, 566780500, 909508600, 936275340, 599999400, 218243025, 34459425;

The row polynomials satisfy P(n,x) = (n+n*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + n*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + n*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 - 1, v[i])))};
T(9)

Triangle begins:
  0:      1;
  1:      1,       1;
  2:      2,       5,        3;
  3:      6,      26,       35,        15;
  4:     24,     154,      340,       315,       105;
  5:    120,    1044,     3304,      4900,      3465,       945;
  6:    720,    8028,    33740,     70532,     78750,     45045,     10395;
  7:   5040,   69264,   367884,   1008980,   1571570,   1406790,    675675,    135135;
  8:  40320,  663696,  4302216,  14777620,  29957620,  37147110,  27747720,  11486475,   2027025;
  9: 362880, 6999840, 53961336, 224655816, 566780500, 909508600, 936275340, 599999400, 218243025, 34459425;

The row polynomials satisfy P(n,x) = (n+n*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + n*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + n*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 + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• 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-c24cc7aceb1eaafd
• 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-c24cc7aceb1eaafd
• 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-c24cc7aceb1eaafd
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-c24cc7aceb1eaafd

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-c24cc7aceb1eaafd root: theorem check

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

Used in 2 proof steps

• e-fa5fc5a793cceca0 obreschkoff_combination → obreschkoff_combination of Coefficient triangle for polynomials used for e.g.f.s for unsi…
• e-fa5fc5a793cceca0 obreschkoff_combination → obreschkoff_combination of Coefficient triangle for polynomials used for e.g.f.s for unsi…

Related sequences ?

No candidate equalities or related sequences.

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