{"id":"n-141ab74a8d7e58b7","revision":1,"first_index":1,"fingerprint":"f6-b9e11a0a9cd0a58e","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","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"],["5","4"],["21","30","10"],["84","168","105","20"],["330","840","756","280","35"],["1287","3960","4620","2520","630","56"],["5005","18018","25740","18480","6930","1260","84"],["19448","80080","135135","120120","60060","16632","2310","120"],["75582","350064","680680","720720","450450","168168","36036","3960","165"],["293930","1511640","3325608","4084080","3063060","1441440","420420","72072","6435","220"],["1144066","6466460","15872220","22170720","19399380","11027016","4084080","960960","135135","10010","286"],["4457400","27457584","74364290","116396280","116396280","77597520","34918884","10501920","2042040","240240","15015","364"],["17383860","115892400","343219800","594914320","669278610","512143632","271591320","99768240","24942060","4084080","408408","21840","455"],["67863915","486748080","1564547400","2974571600","3718214500","3212537328","1963217256","853572720","261891630","55426800","7759752","668304","30940","560"],["265182525","2035917450","7057847160","14602442400","20078358300","19334715400","13385572200","6731030592","2454021570","640179540","116396280","14108640","1058148","42840","680"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","61","377","2241","13073","75517","433905","2485825","14218905","81270333","464387817","2653649025","15167050785","86716873725"],"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":{"denominator":{"coeffs":[["-2"],["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","2"],["2","1"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-1"],["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164793,"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-b5db2586a8a89c95"]},{"id":164794,"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-b5db2586a8a89c95"]},{"id":164795,"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-b5db2586a8a89c95"]},{"id":164796,"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-b5db2586a8a89c95"]}],"region_exact":{},"incoming_edges":[{"id":"e-b5db2586a8a89c95","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-141ab74a8d7e58b7","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":"6b0d08e9b7537d42928d4be5611055bd250b313d9e75e915aa0a4caafc6f12a6","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-a2304b34e998691c","role":"sequence","name":"Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","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-ab7c2ec4d5f3d9f1","kind":"oeis_match","node_a_id":"n-141ab74a8d7e58b7","node_b_id":null,"fingerprint":null,"oeis_id":"A190666","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A190666","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A190666","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-141ab74a8d7e58b7","oeis_id":"A190666","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":16210}],"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:      5,       4;\n   3:     21,      30,      10;\n   4:     84,     168,     105,      20;\n   5:    330,     840,     756,     280,      35;\n   6:   1287,    3960,    4620,    2520,     630,      56;\n   7:   5005,   18018,   25740,   18480,    6930,    1260,     84;\n   8:  19448,   80080,  135135,  120120,   60060,   16632,   2310,   120;\n   9:  75582,  350064,  680680,  720720,  450450,  168168,  36036,  3960,  165;\n  10: 293930, 1511640, 3325608, 4084080, 3063060, 1441440, 420420, 72072, 6435, 220;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n-2)*P(n,x) = (4*n^2+2*n+(3*n^2+n+2)*x)*P(n-1,x) - ((2*n+1)*x+(2*n+1)*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[Together[((4*n^2 + 2*n + (3*n^2 + n + 2)*x)*P[n - 1, x] - ((2*n + 1)*x + (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 + n - 2)]];\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^2 + 2*n + (3*n^2 + n + 2)*x)*v[i - 1] - ((2*n + 1)*x + (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 + n - 2)); 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-a2304b34e998691c","steps":["x \u21a6 x+1"],"edges":["e-b5db2586a8a89c95"],"text":"A062145 \u2192 x \u21a6 x+1"},"headline":"A062145 \u2192 x \u21a6 x+1","chain":null}