A132440 → θ+1 → (θ+1)²

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Submission text ?

Triangle begins:
  0: 0;
  1: 1;
  2: 0, 16;
  3: 0,  0, 81;
  4: 0,  0,  0, 256;
  5: 0,  0,  0,   0, 625;
  6: 0,  0,  0,   0,   0, 1296;
  7: 0,  0,  0,   0,   0,    0, 2401;
  8: 0,  0,  0,   0,   0,    0,    0, 4096;
  9: 0,  0,  0,   0,   0,    0,    0,    0, 6561;

The row polynomials satisfy (n^4-6*n^3+13*n^2-12*n+4)*P(n,x) = (9*n^4-18*n^3+9*n^2)*x^2*P(n-2,x) + (4*n^4-8*n^3+4*n^2)*x^3*d/dx P(n-2,x) + (4*n^4-16*n^3+16*n^2)*x*P(n-1,x) - (4*n^4-16*n^3+16*n^2)*x^2*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 16*x.

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 16*x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^4 - 18*n^3 + 9*n^2)*x^2*P[n - 2, x] + (4*n^4 - 8*n^3 + 4*n^2)*x^3*D[P[n - 2, x], x] + (4*n^4 - 16*n^3 + 16*n^2)*x*P[n - 1, x] - (4*n^4 - 16*n^3 + 16*n^2)*x^2*D[P[n - 1, x], x])/(n^4 - 6*n^3 + 13*n^2 - 12*n + 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 16*x; for(n = 3, N, my(i = n + 1); v[i] = ((9*n^4 - 18*n^3 + 9*n^2)*x^2*v[i - 2] + (4*n^4 - 8*n^3 + 4*n^2)*x^3*deriv(v[i - 2], x) + (4*n^4 - 16*n^3 + 16*n^2)*x*v[i - 1] - (4*n^4 - 16*n^3 + 16*n^2)*x^2*deriv(v[i - 1], x))/(n^4 - 6*n^3 + 13*n^2 - 12*n + 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)

Properties

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

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from theta_plus_one of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix,…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical ope…?
        • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

          • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                • 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 coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Iterated real-rootedness preservers (above)
            • Parent-row provers of polytool polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Veronese sections of Pólya frequency sequences (above)
                • ↑ Parent-row provers of polytool (above)
        • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

          • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Veronese sections of Pólya frequency sequences (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 Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical ope…?
        • ↑ Laguerre’s theorem on differential operators (above)
        • ↑ Laguerre’s theorem on differential operators (above)

Related ?

Candidate equalities ?

• A127648 → (θ+1)² → θ+1 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
• A127773 → θ+1 → reverse
• Hadamard-factor parent of A127773 → (θ+1)²
• A136572 → D → D
• theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
• basis-map coefficients of A026729, shifted → (θ+1)² → θ+1
• A132774 → Veronese 2 → θ+1
• basis-map coefficients of A026729, shifted → (θ+1)² → (θ+1)²
• A136572 → D → θ+1
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• A130123 → D → (θ+1)²
• A038325 → γ
• basis-map coefficients of A297191, shifted → (θ+1)²
• image under θ+1 → (θ+1)² → reverse
• basis-map coefficients of A030528, shifted → θ+1 → D
• A132774 → Veronese 2 → (θ+1)²
• A136572 → D
• image under θ+1 → θ+1 → D
• A133080 → Veronese 2
• veronese parent of Table read by rows: 1 if x = y, 0 otherwise, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
• A127648 → (θ+1)² → D
• basis-map coefficients of A026729 → (θ+1)² → (θ+1)²
• basis-map coefficients of A030528, shifted → D → θ+1
• A136572 → θ+1
• basis-map coefficients of A112227 → θ+1
• A136572 → (θ+1)² → reverse
• A356546 → γ
• A129184: Shift operator, right.
• basis-map coefficients of A297191, shifted → θ+1
• A003506 → γ
• A178821 → γ
• A132440 → θ+1 → D
• A344565 → γ
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
• A130123 → D → θ+1
• A131431 → reverse
• A131431 → (θ+1)² → θ+1
• A127773 → D → θ+1
• A127648 → (θ+1)²
• basis-map coefficients of A112227 → (θ+1)²
• image under θ+1 → D
• basis-map coefficients of A297191, shifted → (θ+1)² → D
• basis-map coefficients of A030528, shifted → reverse
• A127648 → D → θ+1
• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
• basis-map coefficients of A297191, shifted → θ+1 → D
• A132774 → Veronese 2 → D
• basis-map coefficients of A030528, shifted → (θ+1)² → (θ+1)²
• basis-map coefficients of A297191, shifted → (θ+1)² → reverse
• basis-map coefficients of A026729, shifted → (θ+1)² → reverse
• A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
• A127773 → (θ+1)² → reverse
• basis-map coefficients of A026729 → D → θ+1
• A178822 → γ
• basis-map coefficients of A112227 → (θ+1)² → D
• A131431: 3n + 1 preceded by n zeros.
• A136572 → θ+1 → D
• A136572: Triangle read by rows: row n consists of n zeros followed by n!.
• A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
• Hadamard-factor parent of A127773 → D
• A073423: Sums of two powers of zero: triangle read by rows: T(m,n) = 0^n + 0^m, n >= 0, m = 0..n.
• hadamard_factor parent of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
• A127773 → D → D
• A132440 → θ+1
• A130123 → reverse
• basis-map coefficients of A026729, shifted → D → D
• Hadamard-factor parent of A127773 → reverse
• image under θ+1 → (θ+1)²
• basis-map coefficients of A026729, shifted → (θ+1)²
• basis-map coefficients of A297191, shifted → D → D
• A131436 → D
• basis-map coefficients of A026729, shifted → reverse
• A130123 → θ+1 → reverse
• A132774 → Veronese 2 (j=1) → (θ+1)²
• A136572 → (θ+1)² → (θ+1)²
• A129186: Right shift operator generating 1's in shifted spaces.
• basis-map coefficients of A297191, shifted → (θ+1)² → (θ+1)²
• A131436 → reverse
• A131431 → D
• A136572 → (θ+1)² → θ+1
• A038338 → γ
• A178820 → γ
• A127648 → D
• basis-map coefficients of A030528, shifted → θ+1
• A136572 → D → (θ+1)²
• basis-map coefficients of A026729, shifted → θ+1
• A156992 → γ
• basis-map coefficients of A030528, shifted → (θ+1)² → θ+1
• basis-map coefficients of A026729, shifted → θ+1 → D
• A131436 → (θ+1)²
• basis-map coefficients of A297191, shifted → D → θ+1
• basis-map coefficients of A026729 → (θ+1)² → reverse
• image under θ+1 → θ+1 → (θ+1)²
• image under θ+1 → θ+1 → D → θ+1
• A131431 → θ+1
• image under θ+1 → θ+1 → reverse
• basis-map coefficients of A021012 → θ+1 → reverse
• A131436: Triangle read by rows, (n-1) zeros followed by 2^n - 1.
• A132774 → Veronese 2 (j=1) → θ+1
• basis-map coefficients of A021012 → θ+1 → (θ+1)²
• A131431 → θ+1 → D
• A130123 → (θ+1)² → D
• image under θ+1 → reverse
• basis-map coefficients of A021012 → (θ+1)²
• basis-map coefficients of A112227 → (θ+1)² → (θ+1)²
• image under θ+1 → (θ+1)² → D
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
• A131431 → θ+1 → reverse
• basis-map coefficients of A297191, shifted → (θ+1)² → θ+1
• basis-map coefficients of A026729, shifted → θ+1 → reverse
• basis-map coefficients of A112227 → θ+1 → D
• basis-map coefficients of A026729, shifted → D
• basis-map coefficients of A026729 → (θ+1)² → θ+1
• basis-map coefficients of A021012 → θ+1 → D
• basis-map coefficients of A026729 → (θ+1)² → D
• A130123 → (θ+1)² → θ+1
• image under θ+1 → (θ+1)² → θ+1
• A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
• A130123 → D
• basis-map coefficients of A026729 → D → D
• basis-map coefficients of A030528, shifted → (θ+1)² → reverse
• A127773 → reverse
• basis-map coefficients of A030528, shifted → D → D
• A130123 → (θ+1)²
• Hadamard-factor parent of A127773 → θ+1
• A136572 → (θ+1)² → D
• A196347 → γ
• A130123 → (θ+1)² → (θ+1)²
• A132440: Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the clas…
• basis-map coefficients of A030528, shifted → θ+1 → reverse
• basis-map coefficients of A030528, shifted → D
• A132774 → Veronese 2 (j=1)
• A038221 → γ
• image under θ+1 → θ+1 → D → D
• basis-map coefficients of A297191, shifted → D
• basis-map coefficients of Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
• basis-map coefficients of A026729 → D → (θ+1)²
• A131436 → θ+1
• basis-map coefficients of A112227 → D
• A134309 → D
• A130123 → D → D
• A136572 → (θ+1)²
• A127773 → D
• A028326 → γ
• basis-map coefficients of A112227 → (θ+1)² → θ+1
• A127648 → D → D
• basis-map coefficients of A021012 → D
• A127773 → θ+1
• basis-map coefficients of A026729, shifted → (θ+1)² → D
• A131431 → D → θ+1
• basis-map coefficients of A112227 → θ+1 → reverse
• basis-map coefficients of A030528, shifted → (θ+1)²
• basis-map coefficients of A112227 → (θ+1)² → reverse
• A127773 → θ+1 → D
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• A131431 → (θ+1)² → reverse
• basis-map coefficients of A021012 → θ+1
• A131431 → (θ+1)²
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros …
• A085881 → γ
• A132774 → Veronese 2
• A130123 → θ+1 → D
• A131431 → (θ+1)² → (θ+1)²
• A131431 → D → D
• A130123 → θ+1
• A135838 → γ
• A127648 → (θ+1)² → (θ+1)²
• A134309: Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).
• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
• A136521: Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.
• A073424: Triangle read by rows: T(m,n) = parity of 0^n + 0^m, n = 0,1,2,3 ..., m = 0,1,2,3, ... n.
• A131431 → (θ+1)² → D
• A127773 → (θ+1)²
• basis-map coefficients of A030528, shifted → (θ+1)² → D
• basis-map coefficients of A scaled Hermite triangle.
• A132774 → Veronese 2 (j=1) → D
• basis-map coefficients of A021012 → reverse
• A130123 → (θ+1)² → reverse
• basis-map coefficients of A026729, shifted → D → (θ+1)²
• basis-map coefficients of A026729, shifted → D → θ+1
• A136521 → D
• basis-map coefficients of A297191, shifted → D → (θ+1)²
• basis-map coefficients of A026729 → D
• basis-map coefficients of A026729 → (θ+1)²
• basis-map coefficients of A030528, shifted → D → (θ+1)²
• A127648 → D → (θ+1)²

References

1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
3. [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
4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.