{"id":"n-8bbd0ca4e55b3927","revision":0,"first_index":1,"fingerprint":"f6-d6ac9abcd3bcb1be","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["12","12","1"],["120","180","62","1"],["1680","3360","1992","312","1"],["30240","75600","63600","19800","1562","1"],["665280","1995840","2175600","1024800","187572","7812","1"],["17297280","60540480","81073440","51332400","15198120","1735020","39062","1"],["518918400","2075673600","3297127680","2626525440","1085424480","214925760","15849552","195312","1"],["17643225600","79394515200","145919854080","140211751680","74694674880","21461176800","2952530400","143817840","976562","1"],["670442572800","3352212864000","6997614624000","7877181312000","5148576639360","1954889354880","406175431200","39821073600","1300219932","4882812","1"],["28158588057600","154872234316800","361950773184000","467236721952000","362301250671360","171721981791360","48363834214080","7462477360800","530676156120","11731276260","24414062","1"],["1295295050649600","7771770303897600","20103890987980800","29278227154176000","26287849344556800","14971507796044800","5357053897493760","1149965110990080","134291581319280","7016102792160","105727970712","122070312","1"],["64764752532480000","420970891461120000","1194093085170585600","1936832162365900800","1977081282419443200","1316644262251718400","573457127420136960","159225579879125760","26566515794538720","2381182321518000","92266616005200","952284158280","610351562","1"],["3497296636753920000","24481076457277440000","75555855599441817600","135081139652044185600","154539207388940774400","117917912366077824000","60652974476910816000","20770121772041241600","4565102844457995840","600758113472637120","41771832089878800","1208988849650400","8574219533892","3051757812","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","25","363","7345","190803","6056905","227215803","9834640225","482422523043","26448218974585","1602610293394443","106356973179941905","7672110061127082483","597704434705629473065"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-6","-7"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187547,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-a74e64dd050f610d"]},{"id":187548,"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-a74e64dd050f610d"]},{"id":187549,"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-a74e64dd050f610d"]},{"id":187550,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-a74e64dd050f610d"]}],"region_exact":{},"incoming_edges":[{"id":"e-a74e64dd050f610d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-8bbd0ca4e55b3927","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":"ffe649ab3435887c9fa264e299757478dd06ad95e178dd70f45b2efbfdf4fb65","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d023ddeb9da11712","role":"sequence","name":"Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).","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-260a0ed442346426","kind":"fingerprint_related","node_a_id":"n-8bbd0ca4e55b3927","node_b_id":"n-9f974a244ae6ecb1","fingerprint":"f6-d6ac9abcd3bcb1be","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d6ac9abcd3bcb1be"},"other_id":"n-9f974a244ae6ecb1","other":{"id":"n-9f974a244ae6ecb1","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.","first_index":0,"fingerprint":"f6-d6ac9abcd3bcb1be","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["48","48","4"]],"card_rows":[["1"],["4","2"],["48","48","4"],["960","1440","496","8"]],"row_sums_preview":["1","6","100","2904","117520","6105696","387641920","29083622784"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fb16da45c5e69d3d","steps":["x \u21a6 x+1"],"edges":["e-37c20724e7d0c439"],"text":"A257618 \u2192 x \u21a6 x+1"},"headline":"A257618 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-5c0a026eb2e4c421","kind":"fingerprint_related","node_a_id":"n-8bbd0ca4e55b3927","node_b_id":"n-8525597faaa26fd2","fingerprint":"f6-d6ac9abcd3bcb1be","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d6ac9abcd3bcb1be"},"other_id":"n-8525597faaa26fd2","other":{"id":"n-8525597faaa26fd2","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).","first_index":1,"fingerprint":"f6-d6ac9abcd3bcb1be","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","12","12"]],"card_rows":[["1"],["1","2"],["1","12","12"],["1","62","180","120"]],"row_sums_preview":["1","3","25","363","7345","190803","6056905","227215803"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d023ddeb9da11712","steps":["x \u21a6 x/(x+1)"],"edges":["e-da76e5cf7ae9b6e5"],"text":"A142459 \u2192 x \u21a6 x/(x+1)"},"headline":"A142459 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-e9fa15bbd4c1dd59","kind":"fingerprint_related","node_a_id":"n-8bbd0ca4e55b3927","node_b_id":"n-2e3a2796ffe64916","fingerprint":"f6-d6ac9abcd3bcb1be","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d6ac9abcd3bcb1be"},"other_id":"n-2e3a2796ffe64916","other":{"id":"n-2e3a2796ffe64916","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.","first_index":0,"fingerprint":"f6-d6ac9abcd3bcb1be","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["4","48","48"]],"card_rows":[["1"],["2","4"],["4","48","48"],["8","496","1440","960"]],"row_sums_preview":["1","6","100","2904","117520","6105696","387641920","29083622784"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fb16da45c5e69d3d","steps":["x \u21a6 x/(x+1)"],"edges":["e-07e16f43dc58fab2"],"text":"A257618 \u2192 x \u21a6 x/(x+1)"},"headline":"A257618 \u2192 x \u21a6 x/(x+1)","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:           2,           1;\n   3:          12,          12,            1;\n   4:         120,         180,           62,            1;\n   5:        1680,        3360,         1992,          312,           1;\n   6:       30240,       75600,        63600,        19800,        1562,           1;\n   7:      665280,     1995840,      2175600,      1024800,      187572,        7812,          1;\n   8:    17297280,    60540480,     81073440,     51332400,    15198120,     1735020,      39062,         1;\n   9:   518918400,  2075673600,   3297127680,   2626525440,  1085424480,   214925760,   15849552,    195312,      1;\n  10: 17643225600, 79394515200, 145919854080, 140211751680, 74694674880, 21461176800, 2952530400, 143817840, 976562, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n-6+(4*n-7)*x)*P(n-1,x) - (4*x+4*x^2)*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[(4*n - 6 + (4*n - 7)*x)*P[n - 1, x] - (4*x + 4*x^2)*D[P[n - 1, x], x]];\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] = (4*n - 6 + (4*n - 7)*x)*v[i - 1] - (4*x + 4*x^2)*deriv(v[i - 1], x)); 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-d023ddeb9da11712","steps":["x \u21a6 x+1"],"edges":["e-a74e64dd050f610d"],"text":"A142459 \u2192 x \u21a6 x+1"},"headline":"A142459 \u2192 x \u21a6 x+1","chain":null}