A059473 → (θ+1)²

OEIS matches ?

No OEIS matches.

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:   2,     8;
  2:   4,    40,     36;
  3:   8,   128,    288,     128;
  4:  16,   352,   1332,    1408,     400;
  5:  32,   896,   4824,    8576,    5600,    1152;
  6:  64,  2176,  15264,   39040,   42400,   19584,    3136;
  7: 128,  5120,  44352,  149504,  233600,  177408,   62720,    8192;
  8: 256, 11776, 121536,  510976, 1056400, 1149696,  661696,  188416,  20736;
  9: 512, 26624, 319104, 1611776, 4176800, 6014592, 4936064, 2269184, 539136, 51200;

The row polynomials satisfy -(4*n^2+4*n+(18*n^3-8*n^2-24*n)*x+(9*n^4-21*n^3-60*n^2)*x^2-(9*n^4+39*n^3+16*n^2+16*n)*x^3+(6*n^3+8*n^2)*x^4)*P(n,x) = -(8*n^2+8*n+(36*n^3+80*n^2-16*n-64)*x+(18*n^4+138*n^3-308*n-104)*x^2+(36*n^4+48*n^3-284*n^2-392*n-144)*x^3-(78*n^3+242*n^2+212*n+80)*x^4+(24*n^2+56*n+32)*x^5)*P(n-1,x) + ((8*n+8)*x+(36*n^2+96*n+48)*x^2+(18*n^3+210*n^2+316*n+88)*x^3+(72*n^3+288*n^2+244*n+64)*x^4+(18*n^3+18*n^2-60*n-48)*x^5-(12*n^2+28*n+16)*x^6)*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[Together[(-(8*n^2 + 8*n + (36*n^3 + 80*n^2 - 16*n - 64)*x + (18*n^4 + 138*n^3 - 308*n - 104)*x^2 + (36*n^4 + 48*n^3 - 284*n^2 - 392*n - 144)*x^3 - (78*n^3 + 242*n^2 + 212*n + 80)*x^4 + (24*n^2 + 56*n + 32)*x^5)*P[n - 1, x] + ((8*n + 8)*x + (36*n^2 + 96*n + 48)*x^2 + (18*n^3 + 210*n^2 + 316*n + 88)*x^3 + (72*n^3 + 288*n^2 + 244*n + 64)*x^4 + (18*n^3 + 18*n^2 - 60*n - 48)*x^5 - (12*n^2 + 28*n + 16)*x^6)*D[P[n - 1, x], x])/(-4*n^2 - 4*n - (18*n^3 - 8*n^2 - 24*n)*x - (9*n^4 - 21*n^3 - 60*n^2)*x^2 + (9*n^4 + 39*n^3 + 16*n^2 + 16*n)*x^3 - (6*n^3 + 8*n^2)*x^4)]];
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] = (-(8*n^2 + 8*n + (36*n^3 + 80*n^2 - 16*n - 64)*x + (18*n^4 + 138*n^3 - 308*n - 104)*x^2 + (36*n^4 + 48*n^3 - 284*n^2 - 392*n - 144)*x^3 - (78*n^3 + 242*n^2 + 212*n + 80)*x^4 + (24*n^2 + 56*n + 32)*x^5)*v[i - 1] + ((8*n + 8)*x + (36*n^2 + 96*n + 48)*x^2 + (18*n^3 + 210*n^2 + 316*n + 88)*x^3 + (72*n^3 + 288*n^2 + 244*n + 64)*x^4 + (18*n^3 + 18*n^2 - 60*n - 48)*x^5 - (12*n^2 + 28*n + 16)*x^6)*deriv(v[i - 1], x))/(-4*n^2 - 4*n - (18*n^3 - 8*n^2 - 24*n)*x - (9*n^4 - 21*n^3 - 60*n^2)*x^2 + (9*n^4 + 39*n^3 + 16*n^2 + 16*n)*x^3 - (6*n^3 + 8*n^2)*x^4)); 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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, 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-a6e970684438cd5f
• 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-a6e970684438cd5f
• 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-a6e970684438cd5f
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-a6e970684438cd5f

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 0 e-a6e970684438cd5f

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

  • premise (sequence): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, .…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-0ee228386582da48 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

    • preserver_image reversal, shift 0 e-c68c82cb1005adf9

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, .…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang_derivative_lag e-0ee228386582da48 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

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.