{"id":"n-15c46876416f2009","revision":1,"first_index":1,"fingerprint":"f6-ab1b08e67af73826","name":"theta_plus_one_squared of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","8"],["1","24"],["1","48","54"],["1","80","270"],["1","120","810","320"],["1","168","1890","2240"],["1","224","3780","8960","1750"],["1","288","6804","26880","15750"],["1","360","11340","67200","78750","9072"],["1","440","17820","147840","288750","99792"],["1","528","26730","295680","866250","598752","45276"],["1","624","38610","549120","2252250","2594592","588588"],["1","728","54054","960960","5255250","9081072","4120116","219648"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","9","25","103","351","1251","4299","14715","49723","166723","554643","1833217","6023785","19691829"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1","2","0"],["1","-10","0"],["0","12","-1"],["0","-4","3"],["0","0","-3"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-1","-30","-32","0"],["1","6","8","0"],["0","12","59","-8"],["0","-4","-25","-12"],["0","0","-3","0"],["0","0","1","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-4","-16","0"],["0","0","0","0","0"],["0","0","-8","34","-8"],["0","0","0","4","-16"],["0","0","0","2","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163517,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-23d8897c84d2236f"]},{"id":163518,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-23d8897c84d2236f"]},{"id":163519,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-23d8897c84d2236f"]}],"region_exact":{},"incoming_edges":[{"id":"e-23d8897c84d2236f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-15c46876416f2009","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"24c63033b19e1b69cdaee57a35d427f60d5620d1242d0ad70910cc88c0a1d2ff","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-4b168caf0bc63f8b","role":"sequence","name":"T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-901cdd5a3e9a9ade","kind":"fingerprint_related","node_a_id":"n-15c46876416f2009","node_b_id":"n-906c3033a0728ce4","fingerprint":"f6-ab1b08e67af73826","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ab1b08e67af73826"},"other_id":"n-906c3033a0728ce4","other":{"id":"n-906c3033a0728ce4","name":"theta_plus_one_squared of Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).","first_index":0,"fingerprint":"f6-ab1b08e67af73826","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","8"]],"card_rows":[["1"],["1"],["1","8"],["1","24"]],"row_sums_preview":["1","1","9","25","103","351","1251","4299"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75ccd0453a9b6cec","steps":["\u2299 C(n,k)","(\u03b8+1)\u00b2"],"edges":["e-2126fae20f2e5116","e-e1819b2ee0295f94"],"text":"A011973 \u2192 \u2299 C(n,k) \u2192 (\u03b8+1)\u00b2"},"headline":"A011973 \u2192 \u2299 C(n,k) \u2192 (\u03b8+1)\u00b2","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1;\n   3: 1,   8;\n   4: 1,  24;\n   5: 1,  48,   54;\n   6: 1,  80,  270;\n   7: 1, 120,  810,   320;\n   8: 1, 168, 1890,  2240;\n   9: 1, 224, 3780,  8960,  1750;\n  10: 1, 288, 6804, 26880, 15750;\n","recurrence_sentence":"The row polynomials satisfy (n-1-(4*n^3-12*n^2+10*n-2)*x+(n^5-3*n^4+3*n^3-n^2)*x^2)*P(n,x) = (n-1-(4*n^3-12*n^2-6*n+30)*x+(n^5-3*n^4-25*n^3+59*n^2+8*n-32)*x^2+(4*n^5-12*n^3-8*n^2)*x^3)*P(n-1,x) + (2*x-(8*n^2+4)*x^2+(2*n^4+4*n^3+34*n^2-16)*x^3-(8*n^4+16*n^3+8*n^2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 1 - (4*n^3 - 12*n^2 - 6*n + 30)*x + (n^5 - 3*n^4 - 25*n^3 + 59*n^2 + 8*n - 32)*x^2 + (4*n^5 - 12*n^3 - 8*n^2)*x^3)*P[n - 1, x] + (2*x - (8*n^2 + 4)*x^2 + (2*n^4 + 4*n^3 + 34*n^2 - 16)*x^3 - (8*n^4 + 16*n^3 + 8*n^2)*x^4)*D[P[n - 1, x], x])/(n - 1 - (4*n^3 - 12*n^2 + 10*n - 2)*x + (n^5 - 3*n^4 + 3*n^3 - n^2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n - 1 - (4*n^3 - 12*n^2 - 6*n + 30)*x + (n^5 - 3*n^4 - 25*n^3 + 59*n^2 + 8*n - 32)*x^2 + (4*n^5 - 12*n^3 - 8*n^2)*x^3)*v[i - 1] + (2*x - (8*n^2 + 4)*x^2 + (2*n^4 + 4*n^3 + 34*n^2 - 16)*x^3 - (8*n^4 + 16*n^3 + 8*n^2)*x^4)*deriv(v[i - 1], x))/(n - 1 - (4*n^3 - 12*n^2 + 10*n - 2)*x + (n^5 - 3*n^4 + 3*n^3 - n^2)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c914b69838ac70ec","steps":["\u2299 1/k! (shift -1)","(\u03b8+1)\u00b2"],"edges":["e-4d3032ecd7423900","e-23d8897c84d2236f"],"text":"A059344 \u2192 \u2299 1/k! (shift -1) \u2192 (\u03b8+1)\u00b2"},"headline":"A059344 \u2192 \u2299 1/k! (shift -1) \u2192 (\u03b8+1)\u00b2","chain":null}
