A008288 → (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:
   1: 1;
   2: 1,  4;
   3: 1, 12,    9;
   4: 1, 20,   45,   16;
   5: 1, 28,  117,  112,    25;
   6: 1, 36,  225,  400,   225,    36;
   7: 1, 44,  369, 1008,  1025,   396,    49;
   8: 1, 52,  549, 2064,  3225,  2196,   637,   64;
   9: 1, 60,  765, 3696,  8025,  8316,  4165,  960,   81;
  10: 1, 68, 1017, 6032, 17025, 24516, 18473, 7232, 1377, 100;

The row polynomials satisfy -(n^2-n+(6*n^3-20*n^2+15*n-1)*x+(4*n^4-22*n^3+22*n^2+5*n-9)*x^2-(4*n^4+2*n^3-16*n^2+19*n-9)*x^3+(2*n^3-3*n^2+1)*x^4)*P(n,x) = -(n^2-n+(6*n^3-4*n^2-13*n-1)*x+(4*n^4+18*n^3-70*n^2+7*n+23)*x^2+(12*n^4-22*n^3-52*n^2+35*n-15)*x^3-(18*n^3+7*n^2-2*n+7)*x^4+(4*n^2+2*n)*x^5)*P(n-1,x) + (n*x+(6*n^2+5*n-2)*x^2+(4*n^3+40*n^2-8*n-10)*x^3+(24*n^3+28*n^2-20*n+10)*x^4+(4*n^3-8*n^2-9*n+2)*x^5-(2*n^2+n)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (6*n^3 - 4*n^2 - 13*n - 1)*x + (4*n^4 + 18*n^3 - 70*n^2 + 7*n + 23)*x^2 + (12*n^4 - 22*n^3 - 52*n^2 + 35*n - 15)*x^3 - (18*n^3 + 7*n^2 - 2*n + 7)*x^4 + (4*n^2 + 2*n)*x^5)*P[n - 1, x] + (n*x + (6*n^2 + 5*n - 2)*x^2 + (4*n^3 + 40*n^2 - 8*n - 10)*x^3 + (24*n^3 + 28*n^2 - 20*n + 10)*x^4 + (4*n^3 - 8*n^2 - 9*n + 2)*x^5 - (2*n^2 + n)*x^6)*D[P[n - 1, x], x])/(-n^2 + n - (6*n^3 - 20*n^2 + 15*n - 1)*x - (4*n^4 - 22*n^3 + 22*n^2 + 5*n - 9)*x^2 + (4*n^4 + 2*n^3 - 16*n^2 + 19*n - 9)*x^3 - (2*n^3 - 3*n^2 + 1)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n^2 - n + (6*n^3 - 4*n^2 - 13*n - 1)*x + (4*n^4 + 18*n^3 - 70*n^2 + 7*n + 23)*x^2 + (12*n^4 - 22*n^3 - 52*n^2 + 35*n - 15)*x^3 - (18*n^3 + 7*n^2 - 2*n + 7)*x^4 + (4*n^2 + 2*n)*x^5)*v[i - 1] + (n*x + (6*n^2 + 5*n - 2)*x^2 + (4*n^3 + 40*n^2 - 8*n - 10)*x^3 + (24*n^3 + 28*n^2 - 20*n + 10)*x^4 + (4*n^3 - 8*n^2 - 9*n + 2)*x^5 - (2*n^2 + n)*x^6)*deriv(v[i - 1], x))/(-n^2 + n - (6*n^3 - 20*n^2 + 15*n - 1)*x - (4*n^4 - 22*n^3 + 22*n^2 + 5*n - 9)*x^2 + (4*n^4 + 2*n^3 - 16*n^2 + 19*n - 9)*x^3 - (2*n^3 - 3*n^2 + 1)*x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-a12935b47d4fedb1
• Coefficients of one sign. Every nonzero row, n >= 1: 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-a12935b47d4fedb1
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-a12935b47d4fedb1
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-a12935b47d4fedb1

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-a12935b47d4fedb1

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

  • premise (sequence): Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-b176436450506e78 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

    • preserver_image reversal, shift 0 e-73f1cbc14f4fad08

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang_derivative_lag e-b176436450506e78 (expanded above)
        • basis_map A008288 e-796ce5140a0b4b04

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement
        • basis_map A008288 e-b8e0a0fa3ba7bf57

          Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

          • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
            no recorded property covers this requirement
    • ↵ basis_map A008288 e-796ce5140a0b4b04 (expanded above)
    • ↵ basis_map A008288 e-b8e0a0fa3ba7bf57 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• delannoy → (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.