A091044 → ⊙ C(n+2,k+1) → x ↦ x/(x+1)

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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;
  2: 1,   7,     7;
  3: 1,  17,    45,     30;
  4: 1,  34,   180,    292,     146;
  5: 1,  60,   560,   1660,    1930,     772;
  6: 1,  97,  1471,   7079,   14367,   12993,    4331;
  7: 1, 147,  3409,  24885,   78365,  120239,   88893,   25398;
  8: 1, 212,  7168,  75740,  346598,  802116,  986776,  616632,  154158;
  9: 1, 294, 13944, 205740, 1310670, 4287552, 7802928, 8000328, 4328190, 961820;

The row polynomials satisfy (2*n^2+7*n+6+(6*n^2+21*n+18)*x+(6*n^2+21*n+18)*x^2+(2*n^2+7*n+6)*x^3)*P(n,x) = (2*n^2+7*n+6+(22*n^2-n+54)*x+(90*n^2-153*n+198)*x^2+(182*n^2-455*n+378)*x^3+(196*n^2-562*n+396)*x^4+(108*n^2-324*n+216)*x^5+(24*n^2-72*n+48)*x^6)*P(n-1,x) + ((6*n+30)*x-(24*n-216)*x^2-(204*n-660)*x^3-(456*n-1080)*x^4-(474*n-990)*x^5-(240*n-480)*x^6-(48*n-96)*x^7)*d/dx P(n-1,x) + (12*x^2+84*x^3+240*x^4+360*x^5+300*x^6+132*x^7+24*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[((2*n^2 + 7*n + 6 + (22*n^2 - n + 54)*x + (90*n^2 - 153*n + 198)*x^2 + (182*n^2 - 455*n + 378)*x^3 + (196*n^2 - 562*n + 396)*x^4 + (108*n^2 - 324*n + 216)*x^5 + (24*n^2 - 72*n + 48)*x^6)*P[n - 1, x] + ((6*n + 30)*x - (24*n - 216)*x^2 - (204*n - 660)*x^3 - (456*n - 1080)*x^4 - (474*n - 990)*x^5 - (240*n - 480)*x^6 - (48*n - 96)*x^7)*D[P[n - 1, x], x] + (12*x^2 + 84*x^3 + 240*x^4 + 360*x^5 + 300*x^6 + 132*x^7 + 24*x^8)*D[P[n - 1, x], {x, 2}])/(2*n^2 + 7*n + 6 + (6*n^2 + 21*n + 18)*x + (6*n^2 + 21*n + 18)*x^2 + (2*n^2 + 7*n + 6)*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] = ((2*n^2 + 7*n + 6 + (22*n^2 - n + 54)*x + (90*n^2 - 153*n + 198)*x^2 + (182*n^2 - 455*n + 378)*x^3 + (196*n^2 - 562*n + 396)*x^4 + (108*n^2 - 324*n + 216)*x^5 + (24*n^2 - 72*n + 48)*x^6)*v[i - 1] + ((6*n + 30)*x - (24*n - 216)*x^2 - (204*n - 660)*x^3 - (456*n - 1080)*x^4 - (474*n - 990)*x^5 - (240*n - 480)*x^6 - (48*n - 96)*x^7)*deriv(v[i - 1], x) + (12*x^2 + 84*x^3 + 240*x^4 + 360*x^5 + 300*x^6 + 132*x^7 + 24*x^8)*deriv(deriv(v[i - 1], x), x))/(2*n^2 + 7*n + 6 + (6*n^2 + 21*n + 18)*x + (6*n^2 + 21*n + 18)*x^2 + (2*n^2 + 7*n + 6)*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-f3afeb27929af5db
• 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-f3afeb27929af5db
• 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-f3afeb27929af5db

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-f3afeb27929af5db

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image hadamard_multiplier, shift 0 e-00224510dcd0cfb5

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

      • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-4ffbabf9b60188a8 root: theorem check

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

        • preserver_image reversal, shift 0 e-75f779345b39f910

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (expanded above)
    • preserver_image reversal, shift 0 e-3a2967256f8a0d67

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ preserver_image hadamard_multiplier, shift 0 e-00224510dcd0cfb5 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A091044 → ⊙ C(n+2,k+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