A026729 → basis A013609 → basis A013610

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Submission text ?

Triangle begins:
   1:      1;
   2:      3,       6;
   3:     12,      42,       36;
   4:     45,     234,      396,      216;
   5:    171,    1170,     2952,     3240,      1296;
   6:    648,    5508,    18468,    30456,     24624,      7776;
   7:   2457,   24948,   104328,   229608,    279936,    178848,     46656;
   8:   9315,  109998,   551124,  1516968,   2474064,   2387232,   1259712,    279936;
   9:  35316,  475470,  2776032,  9172440,  18741456,  24222240,  19315584,   8678016,  1679616;
  10: 133893, 2024190, 13494276, 52031160, 127783008, 207121536, 221382720, 150325632, 58786560, 10077696;

The row polynomials satisfy P(n,x) = (3+6*x)*P(n-2,x) + (3+6*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 6*x.

P[1, x_] := 1;
P[2, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[(3 + 6*x)*P[n - 2, x] + (3 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 6*x; for(n = 3, N, my(i = n); v[i] = (3 + 6*x)*v[i - 2] + (3 + 6*x)*v[i - 1]); 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)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward ant…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
            • Parent-row provers of polytool polytool ? step › direct check
            • Parent-row provers of polytool polytool ? step › direct check
            • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›

              • from delannoy?
                • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
                • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from delannoy?
                    • ↑ Liu–Wang theorem with a lag row (above)
                    • ↑ Liu–Wang theorem for three-term recurrences (above)
                    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                      • from delannoy?
                        • ↑ Liu–Wang theorem with a lag row (above)
                        • ↑ Liu–Wang theorem for three-term recurrences (above)
                        • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

                          • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
                            no recorded property covers this requirement
                        • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

                          • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
                            no recorded property covers this requirement
                    • ↑ Affine-current basis map theorem (above)
                    • ↑ Affine-current basis map theorem (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • ↑ Affine-current basis map theorem (above)
                • ↑ Affine-current basis map theorem (above)
            • Parent-row provers of polytool polytool ? step › direct check
            • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Geometric basis map (elementary) polytool ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.?
                • Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
                • Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
        • Geometric basis map (elementary) polytool ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            no recorded property covers this requirement
        • ↑ Ma–Wang interval theorem (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Affine-current basis map theorem (above)
        • ↑ Ma–Wang interval theorem (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Affine-current basis map theorem (above)
        • ↑ Ma–Wang interval theorem (above)

Related ?

Same fingerprint ?

• A099093 → basis A013609 → reverse
• A099093 → basis A013609

References

1. [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
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741