A008292 → θ+1 (shift +2)

OEIS matches ?

• A188588: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A357711: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,    8,       3;
   2: 1,   22,      33,        4;
   3: 1,   52,     198,      104,         5;
   4: 1,  114,     906,     1208,       285,         6;
   5: 1,  240,    3573,     9664,      5955,       720,          7;
   6: 1,  494,   12879,    62476,     78095,     25758,       1729,         8;
   7: 1, 1004,   43824,   352936,    780950,    529404,     102256,      4016,        9;
   8: 1, 2026,  143520,  1820768,   6551770,   7862124,    3186344,    382720,     9117,      10;
   9: 1, 4072,  457911,  8813952,  48690570,  94345488,   68166798,  17627904,  1373733,   20360,    11;
  10: 1, 8166, 1434813, 40750740, 331592370, 975073716, 1137586002, 530547792, 91689165, 4782710, 44913, 12;

The row polynomials satisfy P(n,x) = (1+(2*n+2)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) - (n^2+2*n)*x^2*P(n-2,x) - ((n+2)*x^2-(n+2)*x^3)*d/dx P(n-2,x) for n >= 3, with P(1,x) = 1 + 8*x + 3*x^2, P(2,x) = 1 + 22*x + 33*x^2 + 4*x^3.

P[1, x_] := 1 + 8*x + 3*x^2;
P[2, x_] := 1 + 22*x + 33*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n + 2)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] - (n^2 + 2*n)*x^2*P[n - 2, x] - ((n + 2)*x^2 - (n + 2)*x^3)*D[P[n - 2, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 8*x + 3*x^2; v[2] = 1 + 22*x + 33*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = (1 + (2*n + 2)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) - (n^2 + 2*n)*x^2*v[i - 2] - ((n + 2)*x^2 - (n + 2)*x^3)*deriv(v[i - 2], 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)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-fb10cc40ca2d4795
• 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-fb10cc40ca2d4795
• 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-fb10cc40ca2d4795
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-fb10cc40ca2d4795

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

• preserver_image euler_polynomial, shift 2 e-fb10cc40ca2d4795

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): eulerian
    uses: Every row P_n, n >= 3, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-ab3cdcfae289b48d root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers root_domain e-6c46c948823f0a8c root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image reversal, shift 0 e-c61ecf4ef1c45372

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): eulerian
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-ab3cdcfae289b48d (expanded above)
        • ↵ parent_provers root_domain e-6c46c948823f0a8c (expanded above)

Related sequences ?

Same fingerprint (3)

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

• A008292 → θ+1
• A334218 → D
• A173018 → θ+1

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.