delannoy → (θ+1)²

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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,  4;
  2: 1, 12,    9;
  3: 1, 20,   45,   16;
  4: 1, 28,  117,  112,    25;
  5: 1, 36,  225,  400,   225,    36;
  6: 1, 44,  369, 1008,  1025,   396,    49;
  7: 1, 52,  549, 2064,  3225,  2196,   637,   64;
  8: 1, 60,  765, 3696,  8025,  8316,  4165,  960,   81;
  9: 1, 68, 1017, 6032, 17025, 24516, 18473, 7232, 1377, 100;

The row polynomials satisfy -(n^2+n+(6*n^3-2*n^2-7*n)*x+(4*n^4-6*n^3-20*n^2-n)*x^2-(4*n^4+18*n^3+14*n^2+9*n)*x^3+(2*n^3+3*n^2)*x^4)*P(n,x) = -(n^2+n+(6*n^3+14*n^2-3*n-12)*x+(4*n^4+34*n^3+8*n^2-63*n-18)*x^2+(12*n^4+26*n^3-46*n^2-87*n-42)*x^3-(18*n^3+61*n^2+66*n+30)*x^4+(4*n^2+10*n+6)*x^5)*P(n-1,x) + ((n+1)*x+(6*n^2+17*n+9)*x^2+(4*n^3+52*n^2+84*n+26)*x^3+(24*n^3+100*n^2+108*n+42)*x^4+(4*n^3+4*n^2-13*n-11)*x^5-(2*n^2+5*n+3)*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[(-(n^2 + n + (6*n^3 + 14*n^2 - 3*n - 12)*x + (4*n^4 + 34*n^3 + 8*n^2 - 63*n - 18)*x^2 + (12*n^4 + 26*n^3 - 46*n^2 - 87*n - 42)*x^3 - (18*n^3 + 61*n^2 + 66*n + 30)*x^4 + (4*n^2 + 10*n + 6)*x^5)*P[n - 1, x] + ((n + 1)*x + (6*n^2 + 17*n + 9)*x^2 + (4*n^3 + 52*n^2 + 84*n + 26)*x^3 + (24*n^3 + 100*n^2 + 108*n + 42)*x^4 + (4*n^3 + 4*n^2 - 13*n - 11)*x^5 - (2*n^2 + 5*n + 3)*x^6)*D[P[n - 1, x], x])/(-n^2 - n - (6*n^3 - 2*n^2 - 7*n)*x - (4*n^4 - 6*n^3 - 20*n^2 - n)*x^2 + (4*n^4 + 18*n^3 + 14*n^2 + 9*n)*x^3 - (2*n^3 + 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] = (-(n^2 + n + (6*n^3 + 14*n^2 - 3*n - 12)*x + (4*n^4 + 34*n^3 + 8*n^2 - 63*n - 18)*x^2 + (12*n^4 + 26*n^3 - 46*n^2 - 87*n - 42)*x^3 - (18*n^3 + 61*n^2 + 66*n + 30)*x^4 + (4*n^2 + 10*n + 6)*x^5)*v[i - 1] + ((n + 1)*x + (6*n^2 + 17*n + 9)*x^2 + (4*n^3 + 52*n^2 + 84*n + 26)*x^3 + (24*n^3 + 100*n^2 + 108*n + 42)*x^4 + (4*n^3 + 4*n^2 - 13*n - 11)*x^5 - (2*n^2 + 5*n + 3)*x^6)*deriv(v[i - 1], x))/(-n^2 - n - (6*n^3 - 2*n^2 - 7*n)*x - (4*n^4 - 6*n^3 - 20*n^2 - n)*x^2 + (4*n^4 + 18*n^3 + 14*n^2 + 9*n)*x^3 - (2*n^3 + 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-703ce32de304f6f4
• 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-703ce32de304f6f4
• 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-703ce32de304f6f4
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-703ce32de304f6f4

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-703ce32de304f6f4

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] 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 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 (1)

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

• A008288 → (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.