A152198 → (θ+1)²

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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: 1;
  2: 1,  4;
  3: 1,  4;
  4: 1,  8,  9;
  5: 1,  8,  9;
  6: 1, 12, 27, 16;
  7: 1, 12, 27, 16;
  8: 1, 16, 54, 64, 25;
  9: 1, 16, 54, 64, 25;

The row polynomials satisfy (64+(80*n+56)*x+(56*n^2+24*n+4)*x^2+(12*n^3+22*n^2-6*n-12)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (64+(336*n-200)*x+(376*n^2-72*n-220)*x^2+(144*n^3+126*n^2-138*n-116)*x^3+(21*n^4+56*n^3+9*n^2-62*n-20)*x^4+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^5)*P(n-1,x) - (128*x+(272*n+288)*x^2+(160*n^2+552*n+272)*x^3+(32*n^3+276*n^2+412*n+152)*x^4+(2*n^4+44*n^3+142*n^2+156*n+48)*x^5+(2*n^4+12*n^3+26*n^2+24*n+8)*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[((64 + (336*n - 200)*x + (376*n^2 - 72*n - 220)*x^2 + (144*n^3 + 126*n^2 - 138*n - 116)*x^3 + (21*n^4 + 56*n^3 + 9*n^2 - 62*n - 20)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*P[n - 1, x] - (128*x + (272*n + 288)*x^2 + (160*n^2 + 552*n + 272)*x^3 + (32*n^3 + 276*n^2 + 412*n + 152)*x^4 + (2*n^4 + 44*n^3 + 142*n^2 + 156*n + 48)*x^5 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*D[P[n - 1, x], x])/(64 + (80*n + 56)*x + (56*n^2 + 24*n + 4)*x^2 + (12*n^3 + 22*n^2 - 6*n - 12)*x^3 + (n^4 + 2*n^3 + 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] = ((64 + (336*n - 200)*x + (376*n^2 - 72*n - 220)*x^2 + (144*n^3 + 126*n^2 - 138*n - 116)*x^3 + (21*n^4 + 56*n^3 + 9*n^2 - 62*n - 20)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*v[i - 1] - (128*x + (272*n + 288)*x^2 + (160*n^2 + 552*n + 272)*x^3 + (32*n^3 + 276*n^2 + 412*n + 152)*x^4 + (2*n^4 + 44*n^3 + 142*n^2 + 156*n + 48)*x^5 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*deriv(v[i - 1], x))/(64 + (80*n + 56)*x + (56*n^2 + 24*n + 4)*x^2 + (12*n^3 + 22*n^2 - 6*n - 12)*x^3 + (n^4 + 2*n^3 + 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-e247b8d27ee67ca0
• 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-e247b8d27ee67ca0
• 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-e247b8d27ee67ca0

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-e247b8d27ee67ca0

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

  • premise (sequence): Triangle read by rows, A007318 rows repeated.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-3ca43d0d97d5e532

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: triangle of repeated stepped binomial coefficients.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-35a33ce2a6b89415 root: theorem check

          Parent-row provers of polytool polytool ?

    • gamma_polynomial scale 1, shift 0 e-78c582ff5f7b6a70

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-6e3ca670df3d0a74 root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image reversal, shift 0 e-b2e3690a28625635

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-6e3ca670df3d0a74 (expanded above)
    • parent_provers periodic_product e-b274e5cc6dbb8c2d root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

• A274496 → (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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.