{"id":"n-365173a2200e21cc","revision":0,"first_index":0,"fingerprint":"f6-12c97c7d57838f47","name":"shift_x_plus_one of reversal of Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).","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"],["5","1"],["18","10","1"],["56","54","15","1"],["160","224","108","20","1"],["432","800","560","180","25","1"],["1120","2592","2400","1120","270","30","1"],["2816","7840","9072","5600","1960","378","35","1"],["6912","22528","31360","24192","11200","3136","504","40","1"],["16640","62208","101376","94080","54432","20160","4704","648","45","1"],["39424","166400","311040","337920","235200","108864","33600","6720","810","50","1"],["92160","433664","915200","1140480","929280","517440","199584","52800","9240","990","55","1"],["212992","1105920","2601984","3660800","3421440","2230272","1034880","342144","79200","12320","1188","60","1"],["487424","2768896","7188480","11275264","11897600","8895744","4832256","1921920","555984","114400","16016","1404","65","1"],["1105920","6823936","19382272","33546240","39463424","33313280","20756736","9664512","3363360","864864","160160","20384","1638","70","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","29","126","513","1998","7533","27702","99873","354294","1240029","4290894","14703201","49955454","168466797"],"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"],["5","1"],["18","10","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","12","6","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-12","-12","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173980,"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-15dfee1cd172ec35"]},{"id":173981,"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-15dfee1cd172ec35","e-9634b8dccec367f8"]},{"id":173982,"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-15dfee1cd172ec35","e-9634b8dccec367f8"]},{"id":173983,"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-9634b8dccec367f8"]}],"region_exact":{},"incoming_edges":[{"id":"e-9634b8dccec367f8","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-365173a2200e21cc","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":"41c91d431b55ccd4c4b6f792fe0ccc601bb63adfa5c741919cb167b3aaf89fe8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-19ce16e7d575fc90","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).","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-15dfee1cd172ec35","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-365173a2200e21cc","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":"7133d969282a6cd9ba61c7f301e205d308a0799778acfe953b8613a460cd4c74","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-19ce16e7d575fc90","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).","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-b639a85a4c52f56f","kind":"fingerprint_related","node_a_id":"n-8c39fb67b0147cfb","node_b_id":"n-365173a2200e21cc","fingerprint":"f6-12c97c7d57838f47","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-12c97c7d57838f47"},"other_id":"n-8c39fb67b0147cfb","other":{"id":"n-8c39fb67b0147cfb","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-12c97c7d57838f47","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","5"],["1","10","18"]],"card_rows":[["1"],["1","5"],["1","10","18"],["1","15","54","56"]],"row_sums_preview":["1","6","29","126","513","1998","7533","27702"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e7d0ffa461d9dbc4","steps":["x \u21a6 x/(x+1)"],"edges":["e-5fe876e73621d007"],"text":"A125092 \u2192 x \u21a6 x/(x+1)"},"headline":"A125092 \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:     5,     1;\n  2:    18,    10,      1;\n  3:    56,    54,     15,     1;\n  4:   160,   224,    108,    20,     1;\n  5:   432,   800,    560,   180,    25,     1;\n  6:  1120,  2592,   2400,  1120,   270,    30,    1;\n  7:  2816,  7840,   9072,  5600,  1960,   378,   35,   1;\n  8:  6912, 22528,  31360, 24192, 11200,  3136,  504,  40,  1;\n  9: 16640, 62208, 101376, 94080, 54432, 20160, 4704, 648, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+12*x+6*x^2+x^3)*P(n-3,x) - (12+12*x+3*x^2)*P(n-2,x) + (6+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + x, P(2,x) = 18 + 10*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + x;\nP[2, x_] := 18 + 10*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(8 + 12*x + 6*x^2 + x^3)*P[n - 3, x] - (12 + 12*x + 3*x^2)*P[n - 2, x] + (6 + 3*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] = 5 + x; v[3] = 18 + 10*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 12*x + 6*x^2 + x^3)*v[i - 3] - (12 + 12*x + 3*x^2)*v[i - 2] + (6 + 3*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-e7d0ffa461d9dbc4","steps":["reverse","x \u21a6 x+1"],"edges":["e-68e51a2bf17adc8c","e-15dfee1cd172ec35"],"text":"A125092 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A125092 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}