A011973 → (θ+1)²

OEIS matches ?

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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;
  2: 1,  4;
  3: 1,  8;
  4: 1, 12,   9;
  5: 1, 16,  27;
  6: 1, 20,  54,  16;
  7: 1, 24,  90,  64;
  8: 1, 28, 135, 160,  25;
  9: 1, 32, 189, 320, 125;

The row polynomials satisfy (n^2+n+(3*n^3-2*n)*x+(n^4+n^3)*x^2)*P(n,x) = (n^2+n+(3*n^3+8*n^2-2*n-8)*x+(n^4+11*n^3+8*n^2-14*n-4)*x^2+(2*n^4+8*n^3+6*n^2-8*n-8)*x^3)*P(n-1,x) - ((n+1)*x+(3*n^2+10*n+6)*x^2+(n^3+17*n^2+32*n+12)*x^3+(4*n^3+20*n^2+32*n+16)*x^4)*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[((n^2 + n + (3*n^3 + 8*n^2 - 2*n - 8)*x + (n^4 + 11*n^3 + 8*n^2 - 14*n - 4)*x^2 + (2*n^4 + 8*n^3 + 6*n^2 - 8*n - 8)*x^3)*P[n - 1, x] - ((n + 1)*x + (3*n^2 + 10*n + 6)*x^2 + (n^3 + 17*n^2 + 32*n + 12)*x^3 + (4*n^3 + 20*n^2 + 32*n + 16)*x^4)*D[P[n - 1, x], x])/(n^2 + n + (3*n^3 - 2*n)*x + (n^4 + n^3)*x^2)]];
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^2 + n + (3*n^3 + 8*n^2 - 2*n - 8)*x + (n^4 + 11*n^3 + 8*n^2 - 14*n - 4)*x^2 + (2*n^4 + 8*n^3 + 6*n^2 - 8*n - 8)*x^3)*v[i - 1] - ((n + 1)*x + (3*n^2 + 10*n + 6)*x^2 + (n^3 + 17*n^2 + 32*n + 12)*x^3 + (4*n^3 + 20*n^2 + 32*n + 16)*x^4)*deriv(v[i - 1], x))/(n^2 + n + (3*n^3 - 2*n)*x + (n^4 + n^3)*x^2)); 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-268a0da4668d8a40
• 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-268a0da4668d8a40
• 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-268a0da4668d8a40

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-268a0da4668d8a40

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

  • premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers paper_theorem e-0801036cdfff24d2 root: theorem check

      Parent-row provers of polytool polytool ?

    • parent_provers paper_theorem e-0d3631d0f359c9ce root: theorem check

      Parent-row provers of polytool polytool ?

    • gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): delannoy
        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-15c233e4385971d1 root: theorem check

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

        • preserver_image reversal, shift 0 e-266d85ca03a298fa

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): delannoy
            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-15c233e4385971d1 (expanded above)
            • basis_map A008288 e-e771e35dc9ee3fbd

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

              • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                no recorded property covers this requirement
            • basis_map A008288 e-33dba7e79fe9caa8

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

              • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                no recorded property covers this requirement
        • ↵ basis_map A008288 e-e771e35dc9ee3fbd (expanded above)
        • ↵ basis_map A008288 e-33dba7e79fe9caa8 (expanded above)

Related sequences ?

Same fingerprint (2)

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

• A052553 → (θ+1)²
• A169803 → (θ+1)²

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.