A011973 → basis A007318 → (θ+1)²

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No OEIS matches.

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,   4;
  3:  3,   8;
  4:  5,  20,   9;
  5:  8,  40,  27;
  6: 13,  80,  81,  16;
  7: 21, 152, 198,  64;
  8: 34, 284, 459, 224,  25;
  9: 55, 520, 999, 640, 125;

The row polynomials satisfy (100*n+(125*n^2+85*n)*x+(75*n^3+40*n^2+9*n)*x^2+(15*n^4+25*n^3-8*n^2-14*n)*x^3+(n^5+2*n^4+n^3)*x^4)*P(n,x) = (300*n-200+(550*n^2+280*n-620)*x+(300*n^3+760*n^2-268*n-668)*x^2+(55*n^4+375*n^3+266*n^2-394*n-284)*x^3+(3*n^5+54*n^4+125*n^3+8*n^2-142*n-44)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) - (125+(375*n+525)*x+(300*n^2+1250*n+885)*x^2+(75*n^3+815*n^2+1645*n+771)*x^3+(5*n^4+165*n^3+800*n^2+1078*n+378)*x^4+(9*n^4+114*n^3+337*n^2+356*n+108)*x^5+(4*n^4+24*n^3+52*n^2+48*n+16)*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[((300*n - 200 + (550*n^2 + 280*n - 620)*x + (300*n^3 + 760*n^2 - 268*n - 668)*x^2 + (55*n^4 + 375*n^3 + 266*n^2 - 394*n - 284)*x^3 + (3*n^5 + 54*n^4 + 125*n^3 + 8*n^2 - 142*n - 44)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] - (125 + (375*n + 525)*x + (300*n^2 + 1250*n + 885)*x^2 + (75*n^3 + 815*n^2 + 1645*n + 771)*x^3 + (5*n^4 + 165*n^3 + 800*n^2 + 1078*n + 378)*x^4 + (9*n^4 + 114*n^3 + 337*n^2 + 356*n + 108)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(100*n + (125*n^2 + 85*n)*x + (75*n^3 + 40*n^2 + 9*n)*x^2 + (15*n^4 + 25*n^3 - 8*n^2 - 14*n)*x^3 + (n^5 + 2*n^4 + n^3)*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] = ((300*n - 200 + (550*n^2 + 280*n - 620)*x + (300*n^3 + 760*n^2 - 268*n - 668)*x^2 + (55*n^4 + 375*n^3 + 266*n^2 - 394*n - 284)*x^3 + (3*n^5 + 54*n^4 + 125*n^3 + 8*n^2 - 142*n - 44)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] - (125 + (375*n + 525)*x + (300*n^2 + 1250*n + 885)*x^2 + (75*n^3 + 815*n^2 + 1645*n + 771)*x^3 + (5*n^4 + 165*n^3 + 800*n^2 + 1078*n + 378)*x^4 + (9*n^4 + 114*n^3 + 337*n^2 + 356*n + 108)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(100*n + (125*n^2 + 85*n)*x + (75*n^3 + 40*n^2 + 9*n)*x^2 + (15*n^4 + 25*n^3 - 8*n^2 - 14*n)*x^3 + (n^5 + 2*n^4 + n^3)*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-a0717e73efaff775
• 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-a0717e73efaff775
• 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-a0717e73efaff775

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-a0717e73efaff775

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

  • premise (sequence): Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pie…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image affine, shift 0 e-61c0454a07890647

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers paper_theorem e-0801036cdfff24d2 root: theorem check

          Parent-row provers of polytool polytool ?

        • parent_provers paper_theorem e-0d3631d0f359c9ce root: theorem check

          Parent-row provers of polytool polytool ?

        • gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25

          Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

          • 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)
    • preserver_image affine, shift 0 e-ddc26a351d976abe

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers paper_theorem e-0801036cdfff24d2 (expanded above)
        • ↵ parent_provers paper_theorem e-0d3631d0f359c9ce (expanded above)
        • ↵ gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25 (expanded above)
    • basis_map A007318 e-2a29a8b9938e2bd7

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers paper_theorem e-0801036cdfff24d2 (expanded above)
        • ↵ parent_provers paper_theorem e-0d3631d0f359c9ce (expanded above)
        • ↵ gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25 (expanded above)
    • basis_map A007318 e-d0c14addeb793323

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers paper_theorem e-0801036cdfff24d2 (expanded above)
        • ↵ parent_provers paper_theorem e-0d3631d0f359c9ce (expanded above)
        • ↵ gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25 (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. [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.