{"id":"n-2accfa5d21a32af7","revision":0,"first_index":0,"fingerprint":"f6-f9525b8f12f7179b","name":"shift_x_plus_one of basis_map_A007318 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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":true,"rows":[["1"],["3","1"],["21","16","3"],["273","334","135","18"],["5733","9744","6175","1728","180"],["177723","388059","337585","146193","31500","2700"],["7642089","20418720","22665394","13375584","4424553","777600","56700"],["435599073","1377845532","1863651618","1397039320","626715873","168210684","25004700","1587600"],["31798732329","116264290464","185649007266","169075328608","96043674249","34841151360","7880927724","1016064000","57153600"],["2893684641939","12010993387029","22125952732086","23740060230298","16348364144019","7492510114965","2285016234084","447103571580","50923857600","2571912000"],["321198995255229","1492372921266864","3116585389548141","3852074085827808","3120371732652499","1730828650682160","665724858306399","175304389320000","30243244630500","3086294400000","141455160000"],["42719466368945457","219684732215338026","513002469613515777","718020489125275770","669246330107417695","436144744895792214","202776097099773627","67253324427782334","15592441230976500","2406531300813000","222508966680000","9336040560000"],["6706956219924436749","37822621334585815728","97676796842118343017","152743409422522526496","161077271978636088175","120675938697017957808","65855137346536252131","26375307508944169344","7693772578630332552","1594035830243808000","222643349232174000","18821457768960000","728211163680000"],["1227372988246171925067","7531872720242328022383","21316712363554966003359","36840632436954391563315","43376791029539954047161","36741728531610170302789","23033000555844768300501","10819498772671581933873","3808113365203270267320","991841861589977126232","185800993461674370000","23704871551847514000","1846015299928800000","66267215894880000"],["258975700519942276189137","1718099307736979264854848","5288672944335542269058964","10011628242370648050212160","13020769313113141418099046","12307067778271441108840384","8717844613102313993198556","4701379299397404459599808","1939559291188406129461185","609130536141828551703552","143347405087360890324360","24510832210915634304000","2878520741228965770000","207813989046343680000","6958057668962400000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","40","760","23560","1083760","69360640","5895654400","642626329600","87397180825600","14507932017049600","2887078471392870400","678463440777324544000","185898982772986925056000","58744078556263868317696000"],"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"]],"recurrence":{"denominator":{"coeffs":[["2"]]},"terms":[{"coeff":{"coeffs":[["2","0"],["2","1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-5cb59165f26847ca","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["1","1/2"],["1","1/2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190536,"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-70cce12c20508210"]},{"id":190537,"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-70cce12c20508210","e-90df9a6df8f03b45"]},{"id":190538,"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-70cce12c20508210","e-90df9a6df8f03b45"]},{"id":190539,"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-70cce12c20508210","e-90df9a6df8f03b45"]},{"id":190540,"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-90df9a6df8f03b45"]}],"region_exact":{},"incoming_edges":[{"id":"e-90df9a6df8f03b45","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-2accfa5d21a32af7","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":"6087a284ce40499f0fa606fd20e5ff30dd4d27527b6cacb7075f77c247a96854","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-6b8ac0ed9d98a614","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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-5cb59165f26847ca","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-2accfa5d21a32af7","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"e5f9cf739775e90134fe4e055d650716f9272eecff97288bcce6bf25cc1c058e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-70cce12c20508210","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-2accfa5d21a32af7","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":"b8529d213c52a416050f9b68038d777c38b8966abc0fafcff638054204e3e3e9","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-6b8ac0ed9d98a614","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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":[],"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:             3,              1;\n  2:            21,             16,              3;\n  3:           273,            334,            135,             18;\n  4:          5733,           9744,           6175,           1728,            180;\n  5:        177723,         388059,         337585,         146193,          31500,          2700;\n  6:       7642089,       20418720,       22665394,       13375584,        4424553,        777600,         56700;\n  7:     435599073,     1377845532,     1863651618,     1397039320,      626715873,     168210684,      25004700,      1587600;\n  8:   31798732329,   116264290464,   185649007266,   169075328608,    96043674249,   34841151360,    7880927724,   1016064000,    57153600;\n  9: 2893684641939, 12010993387029, 22125952732086, 23740060230298, 16348364144019, 7492510114965, 2285016234084, 447103571580, 50923857600, 2571912000;\n","recurrence_sentence":"The row polynomials satisfy 2*P(n,x) = (2*n^2+2*n+2+(n^2+n)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 2*n + 2 + (n^2 + n)*x)*P[n - 1, x])/(2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 2*n + 2 + (n^2 + n)*x)*v[i - 1])/(2)); 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-ecd77b030b8fab94","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-8bd02eafb0f0a53d","e-70cce12c20508210"],"text":"A128813 \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"A128813 \u2192 basis A007318 \u2192 x \u21a6 x+1","chain":null}