A111910: Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).

OEIS matches ?

• A196148: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A111910: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A111910: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,   1;
  2: 1,   5,     1;
  3: 1,  14,    14,      1;
  4: 1,  30,    84,     30,      1;
  5: 1,  55,   330,    330,     55,      1;
  6: 1,  91,  1001,   2145,   1001,     91,      1;
  7: 1, 140,  2548,  10010,  10010,   2548,    140,     1;
  8: 1, 204,  5712,  37128,  68068,  37128,   5712,   204,   1;
  9: 1, 285, 11628, 116280, 352716, 352716, 116280, 11628, 285, 1;

The row polynomials satisfy (2*n^2+7*n+6)*P(n,x) = -(2*n^2-13*n-4-(16*n^2+n-2)*x)*P(n-1,x) + ((18*n+6)*x-(18*n+6)*x^2)*d/dx P(n-1,x) + (4*n^2-6*n+2+(8*n^2-12*n+4)*x+(4*n^2-6*n+2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 13*n - 4 - (16*n^2 + n - 2)*x)*P[n - 1, x] + ((18*n + 6)*x - (18*n + 6)*x^2)*D[P[n - 1, x], x] + (4*n^2 - 6*n + 2 + (8*n^2 - 12*n + 4)*x + (4*n^2 - 6*n + 2)*x^2)*P[n - 2, x])/(2*n^2 + 7*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 13*n - 4 - (16*n^2 + n - 2)*x)*v[i - 1] + ((18*n + 6)*x - (18*n + 6)*x^2)*deriv(v[i - 1], x) + (4*n^2 - 6*n + 2 + (8*n^2 - 12*n + 4)*x + (4*n^2 - 6*n + 2)*x^2)*v[i - 2])/(2*n^2 + 7*n + 6)); 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)

Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   5,     1;
  3: 1,  14,    14,      1;
  4: 1,  30,    84,     30,      1;
  5: 1,  55,   330,    330,     55,      1;
  6: 1,  91,  1001,   2145,   1001,     91,      1;
  7: 1, 140,  2548,  10010,  10010,   2548,    140,     1;
  8: 1, 204,  5712,  37128,  68068,  37128,   5712,   204,   1;
  9: 1, 285, 11628, 116280, 352716, 352716, 116280, 11628, 285, 1;

The row polynomials satisfy (2*n^2+7*n+6)*P(n,x) = -(2*n^2-13*n-4-(16*n^2+n-2)*x)*P(n-1,x) + ((18*n+6)*x-(18*n+6)*x^2)*d/dx P(n-1,x) + (4*n^2-6*n+2+(8*n^2-12*n+4)*x+(4*n^2-6*n+2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 13*n - 4 - (16*n^2 + n - 2)*x)*P[n - 1, x] + ((18*n + 6)*x - (18*n + 6)*x^2)*D[P[n - 1, x], x] + (4*n^2 - 6*n + 2 + (8*n^2 - 12*n + 4)*x + (4*n^2 - 6*n + 2)*x^2)*P[n - 2, x])/(2*n^2 + 7*n + 6)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 13*n - 4 - (16*n^2 + n - 2)*x)*v[i - 1] + ((18*n + 6)*x - (18*n + 6)*x^2)*deriv(v[i - 1], x) + (4*n^2 - 6*n + 2 + (8*n^2 - 12*n + 4)*x + (4*n^2 - 6*n + 2)*x^2)*v[i - 2])/(2*n^2 + 7*n + 6)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   5,     1;
  3: 1,  14,    14,      1;
  4: 1,  30,    84,     30,      1;
  5: 1,  55,   330,    330,     55,      1;
  6: 1,  91,  1001,   2145,   1001,     91,      1;
  7: 1, 140,  2548,  10010,  10010,   2548,    140,     1;
  8: 1, 204,  5712,  37128,  68068,  37128,   5712,   204,   1;
  9: 1, 285, 11628, 116280, 352716, 352716, 116280, 11628, 285, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n^2+7*n+6)*P(n,x) = -(2*n^2-13*n-4-(16*n^2+n-2)*x)*P(n-1,x) + ((18*n+6)*x-(18*n+6)*x^2)*d/dx P(n-1,x) + (4*n^2-6*n+2+(8*n^2-12*n+4)*x+(4*n^2-6*n+2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 13*n - 4 - (16*n^2 + n - 2)*x)*P[n - 1, x] + ((18*n + 6)*x - (18*n + 6)*x^2)*D[P[n - 1, x], x] + (4*n^2 - 6*n + 2 + (8*n^2 - 12*n + 4)*x + (4*n^2 - 6*n + 2)*x^2)*P[n - 2, x])/(2*n^2 + 7*n + 6)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 13*n - 4 - (16*n^2 + n - 2)*x)*v[i - 1] + ((18*n + 6)*x - (18*n + 6)*x^2)*deriv(v[i - 1], x) + (4*n^2 - 6*n + 2 + (8*n^2 - 12*n + 4)*x + (4*n^2 - 6*n + 2)*x^2)*v[i - 2])/(2*n^2 + 7*n + 6)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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-00224510dcd0cfb5, e-3a2967256f8a0d67
• 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-00224510dcd0cfb5, e-3a2967256f8a0d67
• 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-00224510dcd0cfb5, e-3a2967256f8a0d67

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• definition_equality e-63f6e30cb237e510 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image hadamard_multiplier, shift 0 e-00224510dcd0cfb5

  Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-4ffbabf9b60188a8 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • preserver_image reversal, shift 0 e-75f779345b39f910

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (expanded above)
• preserver_image reversal, shift 0 e-3a2967256f8a0d67

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ preserver_image hadamard_multiplier, shift 0 e-00224510dcd0cfb5 (expanded above)

Used in 7 proof steps

• e-10ee054377433864 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1…
• e-03362dcf8086f907 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((…
• e-3a2967256f8a0d67 preserver_image (reversal, shift 0) → Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1…
• e-69ea95183e485696 preserver_image (derivative_polynomial, shift 0) → derivative of Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+…
• e-b04192f4d27a9c29 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!…
• e-e1d9a8dd959a91f4 preserver_image (affine, shift 0) → shift_x_plus_one of Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1…
• e-f3afeb27929af5db preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1…

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023