A136572 → θ+1 → basis A013611

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Triangle begins:
  0:       1;
  1:       2,         8;
  2:       6,        48,         96;
  3:      24,       288,       1152,        1536;
  4:     120,      1920,      11520,       30720,        30720;
  5:     720,     14400,     115200,      460800,       921600,       737280;
  6:    5040,    120960,    1209600,     6451200,     19353600,     30965760,      20643840;
  7:   40320,   1128960,   13547520,    90316800,    361267200,    867041280,    1156055040,     660602880;
  8:  362880,  11612160,  162570240,  1300561920,   6502809600,  20808990720,   41617981440,   47563407360,   23781703680;
  9: 3628800, 130636800, 2090188800, 19508428800, 117050572800, 468202291200, 1248539443200, 2140353331200, 2140353331200, 951268147200;

The row polynomials satisfy P(n,x) = (n+1+(4*n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (4*n + 4)*x)*P[n - 1, x]];
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 + 1 + (4*n + 4)*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 theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangle read by rows: row n consists of n zeros followed by n!.?
        • 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
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows: row n consists of n zeros followed by n!.?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Parent-row provers of polytool (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: row n consists of n zeros followed by n!.?
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Parent-row provers of polytool (above)
                • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangle read by rows: row n consists of n zeros followed by n!.?
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
        • ↑ Laguerre’s theorem on multiplier sequences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A026729 → D → basis A013611
• basis-map coefficients of A021012 → θ+1 → basis A013611
• basis-map coefficients of A030528, shifted → θ+1 → basis A013611
• A136572 → basis A013611
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013611
• basis-map coefficients of A021012 → basis A013611
• A136521 → basis A013611
• basis-map coefficients of A112227 → θ+1 → basis A013611
• A130123 → basis A013611 → D
• basis-map coefficients of A112227 → (θ+1)² → basis A013611
• basis-map coefficients of A030528, shifted → basis A013611
• basis-map coefficients of A026729 → basis A013611 → reverse
• basis-map coefficients of A026729 → basis A013611
• A130123 → θ+1 → basis A013611
• A038330 → basis A007318 → D
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013611
• A038231 → D → D
• basis-map coefficients of A026729, shifted → D → basis A013611
• A027466 → basis A013609
• A013610 → basis A013611 → x ↦ x/(x+1)
• A131436 → basis A013611
• A127648 → D → basis A013611
• basis-map coefficients of A030528, shifted → D → basis A013611
• A136572 → D → basis A013611
• basis-map coefficients of A112227 → basis A013611 → D
• A132440 → θ+1 → basis A013611
• A131431 → D → basis A013611
• A027467 → basis A013611 → D
• A027466 → basis A013609 → D
• A134309 → basis A013611
• A038330 → basis A007318
• basis-map coefficients of A026729 → (θ+1)² → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• A013611 → D
• A131431 → basis A013611
• A038330 → basis A007318 → reverse
• basis-map coefficients of A026729 → basis A013611 → D
• A136572 → basis A013611 → reverse
• basis-map coefficients of A112227 → basis A013611 → reverse
• A027467 → basis A013611 → reverse
• A120910 → basis A007318 → basis A007318
• image under θ+1 → basis A013611
• A132440 → basis A013611
• A130123 → basis A013611
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• A013611 → D → D
• A131431 → θ+1 → basis A013611
• A038303 → basis A007318 → basis A013610
• A027467 → basis A013611
• A129186 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013611
• A130123 → (θ+1)² → basis A013611
• basis-map coefficients of A026729, shifted → θ+1 → basis A013611
• image under θ+1 → θ+1 → D → basis A013611
• image under θ+1 → (θ+1)² → basis A013611
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• A136572 → (θ+1)² → basis A013611
• A136572 → basis A013611 → D
• A038329 → reverse
• basis-map coefficients of A112227 → basis A013611
• A130123 → D → basis A013611
• A038231 → D
• A038329 → D

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.