A038334 → basis A013609 → basis A013610

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Triangle begins:
  0:               1296,                3456,                 2304;
  1:              46656,              186624,               248832,               110592;
  2:            1679616,             8957952,             17915904,             15925248,               5308416;
  3:           60466176,           403107840,           1074954240,           1433272320,             955514880,             254803968;
  4:         2176782336,         17414258688,          58047528960,         103195607040,          103195607040,           55037657088,           12230590464;
  5:        78364164096,        731398864896,        2925595459584,        6501323243520,         8668430991360,         6934744793088,         3082108796928,          587068342272;
  6:      2821109907456,      30091839012864,      140428582060032,      374476218826752,       624127031377920,       665735500136448,       443823666757632,       169075682574336,        28179280429056;
  7:    101559956668416,    1218719480020992,     6499837226778624,    20221715816644608,     40443431633289216,     53924575511052288,     47932956009824256,     27390260577042432,      9130086859014144,     1352605460594688;
  8:   3656158440062976,   48748779200839680,   292492675205038080,  1039973956284579840,   2426605897997352960,   3882569436795764736,   4313966040884183040,   3286831269245091840,   1643415634622545920,   486937965814087680,    64925062108545024;
  9: 131621703842267136, 1930451656353251328, 12869677709021675520, 51478710836086702080, 137276562229564538880, 256249582828520472576, 341666110438027296768, 325396295655264092160, 216930863770176061440, 96413717231189360640, 25710324594983829504, 3116402981210161152;

The row polynomials satisfy P(n,x) = (36+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 1296 + 3456*x + 2304*x^2.

P[0, x_] := 1296 + 3456*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[(36 + 48*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1296 + 3456*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (36 + 48*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 - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.?
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.?
            • Geometric basis map (elementary) polytool ? step ›

              • from Triangle of coefficients in expansion of (1+5x)^n.?
                • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
                • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
                • Parent-row provers of polytool polytool ? step › direct check
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.?
            • ↑ Geometric basis map (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.?
            • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • Geometric basis map (elementary) polytool ? step ›

              • from basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.?
                • Geometric basis map (elementary) polytool ? step ›

                  • from Triangle of coefficients in expansion of (1+5x)^n.?
                    • ↑ Liu–Wang theorem (above)
                    • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Iterated real-rootedness preservers (above)
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from Triangle of coefficients in expansion of (1+5x)^n.?
                    • ↑ Liu–Wang theorem (above)
                    • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Iterated real-rootedness preservers (above)
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Parent-row provers of polytool polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.?
                • ↑ Geometric basis map (elementary) (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
            • Geometric basis map (elementary) polytool ? step ›

              • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
                • Geometric basis map (elementary) polytool ? step ›

                  • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
                    • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
                    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
                    • Parent-row provers of polytool polytool ? step › direct check
                    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
                    • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                    • ↑ Liu–Wang theorem (above)
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Iterated real-rootedness preservers (above)
            • Geometric basis map (elementary) polytool ? step ›

              • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
                • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
                • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
                • Parent-row provers of polytool polytool ? step › direct check
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A085880 → basis A013611 → basis A013609
• A038279 → basis A013610 → x ↦ x/(x+1)
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013609
• A196347 → x ↦ x+1 → basis A013611
• A038330 → basis A013610 → reverse
• basis-map coefficients of A112227 → basis A013610 → basis A007318
• A156992 → basis A013611 → basis A013609
• A038330 → basis A013610 → D
• A356546 → basis A013611 → basis A013609
• A136572 → basis A013610 → basis A007318
• A038329 → basis A013610 → basis A013609
• A013609 → basis A013609 → D
• A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• A196347 → basis A013611 → basis A013609
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013609
• A038208 → basis A013611 → basis A013609
• basis-map coefficients of A112227 → basis A013609 → basis A013609
• A013610 → basis A013611 → x ↦ x+1
• Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
• A130123 → basis A013609 → basis A013609
• A120909 → basis A013610 → basis A013609
• A038221 → basis A013611 → basis A013609
• basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1
• A038335 → reverse
• A120910 → basis A013611 → D
• A130123 → basis A013610 → x ↦ x+1
• A120910 → D → basis A013611
• A038284 → reverse
• A038329 → basis A013611 → basis A013610
• A013610 → basis A007318
• A135065 → x ↦ x+1 → basis A013611
• A178822 → basis A013611 → basis A013609
• A013609 → D → basis A013609
• A178820 → basis A013611 → basis A013609
• A028326 → basis A013611 → basis A013609
• A178822 → x ↦ x+1 → basis A013611
• A013610 → basis A007318 → D
• A027465 → x ↦ x/(x+1)
• A178821 → basis A007318 → basis A013611
• A028326 → basis A007318 → basis A013611
• basis-map coefficients of A026729 → basis A013610 → basis A007318
• A003506 → x ↦ x+1 → basis A013611
• A013609 → basis A013609
• A038335 → D
• A038330 → basis A013610
• A038330 → x ↦ x/(x+1)
• A120910 → basis A013611
• A038243 → basis A013609 → x ↦ x/(x+1)
• A178821 → basis A013611 → basis A013609
• A178820 → basis A007318 → basis A013611
• A120910 → basis A013611 → reverse
• A085880 → x ↦ x+1 → basis A013611
• A027465 → D → basis A013610
• basis-map coefficients of A026729 → basis A013609 → basis A013609
• A085881 → basis A013611 → basis A013609
• A038284 → D
• A094305 → x ↦ x+1 → basis A013611
• A356546 → x ↦ x+1 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.