{"id":"n-a324551c9e9baae3","revision":0,"first_index":0,"fingerprint":"f6-dc0dfc2a3d33df11","name":"shift_x_plus_one of veronese_section_2_1 of A Riordan array of coordination sequences.","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"],["4"],["9","1"],["20","8"],["49","34","1"],["120","112","12"],["289","339","75","1"],["696","992","340","16"],["1681","2820","1302","132","1"],["4060","7824","4548","768","20"],["9801","21333","14994","3626","205","1"],["23660","57416","47388","15024","1460","24"],["57121","152902","144975","57108","8255","294","1"],["137904","403616","432392","204160","39984","2480","28"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","4","10","28","84","244","704","2044","5936","17220","49960","144972","420656","1220564"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["4"],["9","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172788,"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-8aa98efe385635fc"]},{"id":172789,"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-8aa98efe385635fc"]},{"id":172790,"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-8aa98efe385635fc"]}],"region_exact":{},"incoming_edges":[{"id":"e-8aa98efe385635fc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-a324551c9e9baae3","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":"9f581339e168261f1116f0e7f8a801b5a4b2ea56ee86306de79a80c1faad9863","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-c2a4f525057473eb","role":"sequence","name":"veronese_section_2_1 of A Riordan array of coordination sequences.","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:    0;\n  1:    1;\n  2:    4;\n  3:    9,    1;\n  4:   20,    8;\n  5:   49,   34,    1;\n  6:  120,  112,   12;\n  7:  289,  339,   75,   1;\n  8:  696,  992,  340,  16;\n  9: 1681, 2820, 1302, 132, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-4,x) + (2+2*x)*P(n-3,x) + x*P(n-2,x) + 2*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 4, P(3,x) = 9 + x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 4;\nP[3, x_] := 9 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 4, x] + (2 + 2*x)*P[n - 3, x] + x*P[n - 2, x] + 2*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 4; v[4] = 9 + x; for(n = 4, N, my(i = n + 1); v[i] = (1 + x)*v[i - 4] + (2 + 2*x)*v[i - 3] + x*v[i - 2] + 2*v[i - 1]); 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-ec35d1145e9a1211","steps":["Veronese 2 (j=1)","x \u21a6 x+1"],"edges":["e-af7260029131ec6a","e-8aa98efe385635fc"],"text":"A113413 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1"},"headline":"A113413 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1","chain":null}