A013618 → D

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Submission text ?

Triangle begins:
  0:  22,   242;
  1:  33,   726,   3993;
  2:  44,  1452,  15972,    58564;
  3:  55,  2420,  39930,   292820,    805255;
  4:  66,  3630,  79860,   878460,   4831530,   10629366;
  5:  77,  5082, 139755,  2049740,  16910355,   74405562,   136410197;
  6:  88,  6776, 223608,  4099480,  45094280,  297622248,  1091281576,   1714871048;
  7:  99,  8712, 335412,  7379064, 101462130,  892866744,  4910767092,  15433839432,   21221529219;
  8: 110, 10890, 479160, 12298440, 202924260, 2232166860, 16369223640,  77169197160,  212215292190,  259374246010;
  9: 121, 13310, 658845, 19326120, 372027810, 4910767092, 45015365010, 282953722920, 1167184107045, 2853116706110, 3138428376721;

The row polynomials satisfy (n+1)*P(n,x) = (n+2+(11*n+22)*x)*P(n-1,x) for n >= 1, with P(0,x) = 22 + 242*x.

P[0, x_] := 22 + 242*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (11*n + 22)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 22 + 242*x; for(n = 1, N, my(i = n + 1); v[i] = ((n + 2 + (11*n + 22)*x)*v[i - 1])/(n + 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)

Properties

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

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from Triangle of coefficients in expansion of (1+11x)^n.?
    • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    • 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

Related ?

Same fingerprint ?

• A013618 → reverse
• Triangle of coefficients in expansion of (1+11x)^n.
• A038303 → basis A007318 → reverse
• A038303 → basis A007318
• A038303 → basis A007318 → D

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.