{"id":"n-9cb23cc1a7a54971","revision":0,"first_index":0,"fingerprint":"f6-dcf9dad318e02ab9","name":"shift_x_plus_one of veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3"],["13","4"],["63","36"],["313","248","16"],["1563","1560","240"],["7813","9372","2352","64"],["39063","54684","19152","1344"],["195313","312496","140640","17152","256"],["976563","1757808","966816","172800","6912"],["4882813","9765620","6347680","1513600","108800","1024"],["24414063","53710932","40283232","12084864","1309440","33792"],["122070313","292968744","249023472","90331904","13367040","632832","4096"],["610351563","1586914056","1507568400","642700032","121994496","8865792","159744"],["3051757813","8544921868","8972168016","4400634560","1026459392","103498752","3469312","16384"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","3","17","99","577","3363","19601","114243","665857","3880899","22619537","131836323","768398401","4478554083","26102926097"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3"]],"recurrence":{"denominator":{"coeffs":[["-3"],["3"]]},"terms":[{"coeff":{"coeffs":[["-13","-4"],["13","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["10","2","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-dfa50262c79d2eb1","recurrence":{"first_index":0,"initial_polynomials":[["1"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-5","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178003,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-e463472b22b3469a"]},{"id":178004,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-e463472b22b3469a","e-fdde52308f624e42"]},{"id":178005,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-e463472b22b3469a","e-fdde52308f624e42"]},{"id":178006,"position":3,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-fdde52308f624e42"]},{"id":178007,"position":4,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-fdde52308f624e42"]}],"region_exact":{},"incoming_edges":[{"id":"e-fdde52308f624e42","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-9cb23cc1a7a54971","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"cea8cd2bbdad4dbf29145f7d1131e298bef2f2382712427d592aa8f940268e12","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5fccad4e67ef9278","role":"sequence","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-dfa50262c79d2eb1","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-9cb23cc1a7a54971","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"440f9336ac5c28cb983b0e38613014533812898c8850dc3896bbc31a85d28781","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-e463472b22b3469a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-9cb23cc1a7a54971","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"1e50c70def2a35399ce991171ac99dc0b050412f5200713d42bccdda12880327","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-5fccad4e67ef9278","role":"sequence","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      3;\n  2:     13,       4;\n  3:     63,      36;\n  4:    313,     248,     16;\n  5:   1563,    1560,    240;\n  6:   7813,    9372,   2352,     64;\n  7:  39063,   54684,  19152,   1344;\n  8: 195313,  312496, 140640,  17152,  256;\n  9: 976563, 1757808, 966816, 172800, 6912;\n","recurrence_sentence":"The row polynomials satisfy (3*n-3)*P(n,x) = (13*n-13+(4*n-4)*x)*P(n-1,x) + (10+2*x-8*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((13*n - 13 + (4*n - 4)*x)*P[n - 1, x] + (10 + 2*x - 8*x^2)*D[P[n - 1, x], x])/(3*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((13*n - 13 + (4*n - 4)*x)*v[i - 1] + (10 + 2*x - 8*x^2)*deriv(v[i - 1], x))/(3*n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-01cea33b6485d9cc","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-4fd32b1a93a66d21","e-e463472b22b3469a"],"text":"A038220 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A038220 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}