A143491 → basis A008277 → x ↦ x+1

Matches ?

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Submission text ?

Triangle begins:
  0:          21,          10,           1;
  1:         136,          93,          18,           1;
  2:        1045,         916,         258,          28,          1;
  3:        9276,        9805,        3580,         570,         40,          1;
  4:       93289,      114486,       50895,       10580,       1095,         54,         1;
  5:     1047376,     1454425,      756966,      192815,      26180,       1911,        70,       1;
  6:    12975561,    20014408,    11873428,     3561096,     595070,      57176,      3108,      88,      1;
  7:   175721140,   296909721,   196925688,    67670148,   13368096,    1585710,    113736,    4788,    108,     1;
  8:  2581284541,  4726308610,  3453805485,  1333120440,  302856330,   42593292,   3780210,  210360,   7065,   130,   1;
  9: 40864292184, 80383524661, 63986557690, 27328278285, 6997500840, 1134810138, 119670012, 8254290, 366960, 10065, 154, 1;

The row polynomials satisfy P(n,x) = (n+5+x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 21 + 10*x + x^2.

P[0, x_] := 21 + 10*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n + 5 + x)*P[n - 1, x] + (1 + x)*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] = 21 + 10*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 5 + x)*v[i - 1] + (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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A008277 of Unsigned 2-Stirling numbers of the first kind.?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A136124 → basis A008277 → x ↦ x+1

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [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
4. [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
5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.