A108838 → ⊙ C(n+1,k+1) (shift +1) → θ+1

OEIS matches ?

• A350265: 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:
  0: 1;
  1: 1,   2;
  2: 1,   8,     3;
  3: 1,  20,    30,      4;
  4: 1,  40,   150,     80,      5;
  5: 1,  70,   525,    700,    175,      6;
  6: 1, 112,  1470,   3920,   2450,    336,      7;
  7: 1, 168,  3528,  16464,  20580,   7056,    588,     8;
  8: 1, 240,  7560,  56448, 123480,  84672,  17640,   960,    9;
  9: 1, 330, 14850, 166320, 582120, 698544, 291060, 39600, 1485, 10;

The row polynomials satisfy (n^5+10*n^4+35*n^3+50*n^2+24*n)*P(n,x) = (n^5+10*n^4+35*n^3+50*n^2+24*n+(10*n^5+55*n^4+98*n^3+65*n^2+12*n)*x)*P(n-1,x) + ((3*n^4+33*n^3+108*n^2+123*n+36)*x-(15*n^4+69*n^3+87*n^2+24*n)*x^2)*d/dx P(n-1,x) + ((6*n^3+28*n^2+37*n+12)*x^2+(6*n^3+28*n^2+37*n+12)*x^3)*d^2/dx^2 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^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (10*n^5 + 55*n^4 + 98*n^3 + 65*n^2 + 12*n)*x)*P[n - 1, x] + ((3*n^4 + 33*n^3 + 108*n^2 + 123*n + 36)*x - (15*n^4 + 69*n^3 + 87*n^2 + 24*n)*x^2)*D[P[n - 1, x], x] + ((6*n^3 + 28*n^2 + 37*n + 12)*x^2 + (6*n^3 + 28*n^2 + 37*n + 12)*x^3)*D[P[n - 1, x], {x, 2}])/(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)]];
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^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (10*n^5 + 55*n^4 + 98*n^3 + 65*n^2 + 12*n)*x)*v[i - 1] + ((3*n^4 + 33*n^3 + 108*n^2 + 123*n + 36)*x - (15*n^4 + 69*n^3 + 87*n^2 + 24*n)*x^2)*deriv(v[i - 1], x) + ((6*n^3 + 28*n^2 + 37*n + 12)*x^2 + (6*n^3 + 28*n^2 + 37*n + 12)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)); 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-f6a16136bb4575f0
• 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-f6a16136bb4575f0
• 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-f6a16136bb4575f0

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-f6a16136bb4575f0

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

  • premise (sequence): Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <=…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image hadamard_multiplier, shift 1 e-548c39a476f786f5

      Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

      • premise (sequence): Triangle of Dyck paths counted by number of long interior inclines.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-a4549745e8cd4154 root: theorem check

          Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • preserver_image reversal, shift 0 e-97ff2ea1491caf9e

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <=…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ preserver_image hadamard_multiplier, shift 1 e-548c39a476f786f5 (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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023