A108838 → ⊙ C(n+1,k+1) (shift +1) → x ↦ x/(x+1)

OEIS matches ?

• A368708: 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,   6,    6;
  3: 1,  13,   33,    22;
  4: 1,  24,  116,   184,     92;
  5: 1,  40,  325,   920,   1055,     422;
  6: 1,  62,  785,  3520,   6945,    6222,    2074;
  7: 1,  91, 1701, 11291,  34055,   51261,   37639,   10754;
  8: 1, 128, 3388, 31808, 137326,  308672,  374500,  232808,   58202;
  9: 1, 174, 6306, 80934, 479220, 1507170, 2685690, 2724780, 1468080, 326240;

The row polynomials satisfy (n^2+7*n+12+(3*n^2+21*n+36)*x+(3*n^2+21*n+36)*x^2+(n^2+7*n+12)*x^3)*P(n,x) = (n^2+7*n+12+(11*n^2+17*n+72)*x+(45*n^2-45*n+180)*x^2+(91*n^2-203*n+252)*x^3+(98*n^2-274*n+216)*x^4+(54*n^2-162*n+108)*x^5+(12*n^2-36*n+24)*x^6)*P(n-1,x) + ((3*n+24)*x-(12*n-144)*x^2-(102*n-384)*x^3-(228*n-576)*x^4-(237*n-504)*x^5-(120*n-240)*x^6-(24*n-48)*x^7)*d/dx P(n-1,x) + (6*x^2+42*x^3+120*x^4+180*x^5+150*x^6+66*x^7+12*x^8)*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^2 + 7*n + 12 + (11*n^2 + 17*n + 72)*x + (45*n^2 - 45*n + 180)*x^2 + (91*n^2 - 203*n + 252)*x^3 + (98*n^2 - 274*n + 216)*x^4 + (54*n^2 - 162*n + 108)*x^5 + (12*n^2 - 36*n + 24)*x^6)*P[n - 1, x] + ((3*n + 24)*x - (12*n - 144)*x^2 - (102*n - 384)*x^3 - (228*n - 576)*x^4 - (237*n - 504)*x^5 - (120*n - 240)*x^6 - (24*n - 48)*x^7)*D[P[n - 1, x], x] + (6*x^2 + 42*x^3 + 120*x^4 + 180*x^5 + 150*x^6 + 66*x^7 + 12*x^8)*D[P[n - 1, x], {x, 2}])/(n^2 + 7*n + 12 + (3*n^2 + 21*n + 36)*x + (3*n^2 + 21*n + 36)*x^2 + (n^2 + 7*n + 12)*x^3)]];
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 + 7*n + 12 + (11*n^2 + 17*n + 72)*x + (45*n^2 - 45*n + 180)*x^2 + (91*n^2 - 203*n + 252)*x^3 + (98*n^2 - 274*n + 216)*x^4 + (54*n^2 - 162*n + 108)*x^5 + (12*n^2 - 36*n + 24)*x^6)*v[i - 1] + ((3*n + 24)*x - (12*n - 144)*x^2 - (102*n - 384)*x^3 - (228*n - 576)*x^4 - (237*n - 504)*x^5 - (120*n - 240)*x^6 - (24*n - 48)*x^7)*deriv(v[i - 1], x) + (6*x^2 + 42*x^3 + 120*x^4 + 180*x^5 + 150*x^6 + 66*x^7 + 12*x^8)*deriv(deriv(v[i - 1], x), x))/(n^2 + 7*n + 12 + (3*n^2 + 21*n + 36)*x + (3*n^2 + 21*n + 36)*x^2 + (n^2 + 7*n + 12)*x^3)); 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 [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-a929acd24243c7cc
• 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-a929acd24243c7cc
• 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-a929acd24243c7cc

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 mobius, shift 0 e-a929acd24243c7cc

  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

      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 ?

Same fingerprint (1)

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

• A108838 → ⊙ C(n+1,k+1) (shift +1) → x ↦ x+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. [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