{"id":"n-27fa2a3cd638c61f","revision":0,"first_index":0,"fingerprint":"f6-783f05b943b24c5b","name":"shift_x_plus_one of reversal of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","1"],["11","9","1"],["38","49","15","1"],["131","236","129","22","1"],["490","1125","878","269","30","1"],["1897","5309","5436","2473","490","39","1"],["7714","25435","32205","19558","5845","816","49","1"],["32501","123438","186550","142502","57915","12238","1274","60","1"],["142142","610867","1072938","989184","512362","149001","23436","1894","72","1"],["641795","3079205","6166611","6670878","4218495","1582047","344505","41898","2709","85","1"],["2989294","15836313","35618087","44253831","33126420","15318945","4340301","732261","70908","3755","99","1"],["14321923","83034920","207268095","291013106","252068520","139171356","49044723","10828062","1453968","114740","5071","114","1"],["70493962","443950261","1217932710","1907501159","1878473660","1208646342","512490246","141716715","24980364","2728730","178838","6699","130","1"],["355816769","2418747129","7234360906","12507153043","13810672869","10164364484","5056868019","1694015451","375907893","53968444","4883571","270011","8684","147","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","21","103","519","2793","15645","91623","556479","3501897","22748229","152290215","1048324599","7409099817","53677037421"],"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"],["4","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"],["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175367,"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-801d51cbd5c5cf9e"]},{"id":175368,"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-42f2bfe28af837c8","e-801d51cbd5c5cf9e"]},{"id":175369,"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-42f2bfe28af837c8","e-801d51cbd5c5cf9e"]},{"id":175370,"position":3,"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-42f2bfe28af837c8","e-801d51cbd5c5cf9e"]},{"id":175371,"position":4,"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-42f2bfe28af837c8"]}],"region_exact":{},"incoming_edges":[{"id":"e-42f2bfe28af837c8","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-27fa2a3cd638c61f","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":"83b78d9cc741a0766fb8ecae4c06d85c6962062a557c7311edbc76fab6d2957b","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-3706be5862de5f25","role":"sequence","name":"reversal of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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-801d51cbd5c5cf9e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-27fa2a3cd638c61f","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":"43d2efe3775b418f56f513998f9970e0697d3e71526035257ccdd80e93042056","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3706be5862de5f25","role":"sequence","name":"reversal of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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":[{"id":"l-8d1e9c0d0dd061eb","kind":"fingerprint_related","node_a_id":"n-3659b3de32876d6c","node_b_id":"n-27fa2a3cd638c61f","fingerprint":"f6-783f05b943b24c5b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-783f05b943b24c5b"},"other_id":"n-3659b3de32876d6c","other":{"id":"n-3659b3de32876d6c","name":"mobius_x_over_x_plus_one of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","first_index":0,"fingerprint":"f6-783f05b943b24c5b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","9","11"]],"card_rows":[["1"],["1","4"],["1","9","11"],["1","15","49","38"]],"row_sums_preview":["1","5","21","103","519","2793","15645","91623"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fa21c8779d2bdd1","steps":["x \u21a6 x/(x+1)"],"edges":["e-8d9c6b636eca438b"],"text":"A249248 \u2192 x \u21a6 x/(x+1)"},"headline":"A249248 \u2192 x \u21a6 x/(x+1)","chain":null}}],"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:      4,      1;\n  2:     11,      9,       1;\n  3:     38,     49,      15,      1;\n  4:    131,    236,     129,     22,      1;\n  5:    490,   1125,     878,    269,     30,      1;\n  6:   1897,   5309,    5436,   2473,    490,     39,     1;\n  7:   7714,  25435,   32205,  19558,   5845,    816,    49,    1;\n  8:  32501, 123438,  186550, 142502,  57915,  12238,  1274,   60,  1;\n  9: 142142, 610867, 1072938, 989184, 512362, 149001, 23436, 1894, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+1+(n+1)*x)*P(n-2,x) + (2+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[(n + 1 + (n + 1)*x)*P[n - 2, x] + (2 + x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (n + 1)*x)*v[i - 2] + (2 + x)*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-5fa21c8779d2bdd1","steps":["reverse","basis A007318"],"edges":["e-6f752e4b71849668","e-42f2bfe28af837c8"],"text":"A249248 \u2192 reverse \u2192 basis A007318"},"headline":"A249248 \u2192 reverse \u2192 basis A007318","chain":null}