{"id":"n-c8d7801a5ac304d1","revision":0,"first_index":1,"fingerprint":"f6-9675e676835fe620","name":"shift_x_plus_one of Triangle of Ward numbers T(n,k) read by rows.","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"],["4","3"],["26","40","15"],["236","550","420","105"],["2752","8596","9940","5040","945"],["39208","153580","238168","182700","69300","10395"],["660032","3109296","6050940","6224680","3568950","1081080","135135"],["12818912","70564896","165256080","213380860","164023860","75044970","18918900","2027025"],["282137824","1777121984","4865852376","7562881040","7296949660","4474530060","1702701000","367567200","34459425"],["6939897856","49223755328","154288092816","280452180008","325752427000","250698968520","127819451760","41626985400","7856748900","654729075"],["188666182784","1488011230784","5254095662816","10936000675600","14857583060320","13765443371680","8807063265000","3841775617680","1093397555250","183324141000","13749310575"],["5617349020544","48765233950784","191530074540896","449200114963600","698927604094720","757461528984160","583388863817760","319290709537800","121684453991100","30752624652750","4638100767300","316234143225"],["181790703209728","1722537395565312","7448810764402176","19436704784538496","34082062554780240","42305209109042880","38113671490214320","25109194083573600","12004345849696200","4061430238565700","922982052692700","126493657290000","7905853580625"],["6353726042486272","65250314121495552","308054663853123840","885203903046541312","1727210528745968448","2416384728803725680","2493497730018174880","1921496815981863120","1105681848120649800","469201095855191700","142714360609821000","29462481010462500","3699939475732500","213458046676875"],["238513970965257728","2638884554522186752","13505736026966520576","42377969658534050816","91071826779554626720","141790494656856689280","164917455279260443760","145564738637641042720","97973002265922944160","50035499784720450000","19085776927271554500","5272495973796051000","997133688709908750","115551955934415000","6190283353629375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","81","1311","27273","693351","20830113","722035503","28364200569","1245313236663","60429110073489","3211572892464639","185523138537151977","11574425593731913479","775591270444009771137"],"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":[["2","1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","3","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186503,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-2, -1)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-2, -1)","region_exact":null,"edges":["e-7e651bb689988219"]},{"id":186504,"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-7e651bb689988219"]},{"id":186505,"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-7e651bb689988219"]},{"id":186506,"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-7e651bb689988219"]}],"region_exact":{},"incoming_edges":[{"id":"e-7e651bb689988219","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c8d7801a5ac304d1","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":"b51f950f7290a71a89a34d157f0fb0235340e053ff90ec380278ccf176c0c1c3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3fea9ad1525b48eb","role":"sequence","name":"Triangle of Ward numbers T(n,k) read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 0)"}]}],"used_by":[],"used_by_total":0,"links":[],"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:          4,           3;\n   3:         26,          40,           15;\n   4:        236,         550,          420,          105;\n   5:       2752,        8596,         9940,         5040,          945;\n   6:      39208,      153580,       238168,       182700,        69300,        10395;\n   7:     660032,     3109296,      6050940,      6224680,      3568950,      1081080,       135135;\n   8:   12818912,    70564896,    165256080,    213380860,    164023860,     75044970,     18918900,     2027025;\n   9:  282137824,  1777121984,   4865852376,   7562881040,   7296949660,   4474530060,   1702701000,   367567200,   34459425;\n  10: 6939897856, 49223755328, 154288092816, 280452180008, 325752427000, 250698968520, 127819451760, 41626985400, 7856748900, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+2+(n+1)*x)*P(n-1,x) + (2+3*x+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[(n + 2 + (n + 1)*x)*P[n - 1, x] + (2 + 3*x + 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] = (n + 2 + (n + 1)*x)*v[i - 1] + (2 + 3*x + 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-3fea9ad1525b48eb","steps":["x \u21a6 x+1"],"edges":["e-7e651bb689988219"],"text":"A134991 \u2192 x \u21a6 x+1"},"headline":"A134991 \u2192 x \u21a6 x+1","chain":null}
