{"id":"n-e0df8ce9d576df35","revision":0,"first_index":0,"fingerprint":"f6-38173b80df527fb6","name":"shift_x_plus_one of basis_map_A013611 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 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":true,"rows":[["1"],["10","8"],["110","168","64"],["1320","2896","2112","512"],["17160","48208","50624","23552","4096"],["240240","812192","1094400","734720","245760","32768"],["3603600","14104800","22913536","19776000","9564160","2457600","262144"],["57657600","254505600","479454976","499724288","311234560","115834880","23855104","2097152"],["980179200","4787856000","10186779392","12330952704","9288781824","4459069440","1332215808","226492416","16777216"],["17643225600","94022841600","221664877056","303451383808","265845694464","154573504512","59652440064","14734589952","2113929216","134217728"],["335221286400","1927579795200","4963815396864","7538895308800","7478679265280","5063662141440","2369984397312","757176729600","158041374720","19461570560","1073741824"],["6704425728000","41233366195200","114696946298880","190488429350912","209884747776000","161102676951040","87908985077760","34103409770496","9218241331200","1653562408960","177167400960","8589934592"],["140792940288000","919536095923200","2738502801838080","4917832586760192","5931487138103296","5062234198179840","3134910102241280","1419443485802496","466410346119168","108470741237760","16949014691840","1597727834112","68719476736"],["3097444686336000","21356137632614400","67603350407823360","130100339323428864","169835377732354048","158821049464782848","109465895834746880","56307037505585152","21616575501041664","6117639076184064","1240644253122560","170742129885184","14293651161088","549755813888"],["71241227785728000","515970723040819200","1725726160440852480","3533134607701450752","4947016402431574016","5011567159548837888","3788283999917441024","2170789029306433536","947637536568639488","313638302760566784","77475930431291392","13852223012339712","1694691015786496","126993593008128","4398046511104"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","342","6840","143640","3160080","72681840","1744364160","43609104000","1133836704000","30613591008000","857180548224000","24858235898496000","745747076954880000","23118159385601280000"],"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":{"terms":[{"coeff":{"coeffs":[["9","8"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187741,"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-cafb3932c879a6b8"]},{"id":187742,"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-679681cb5c84444a","e-cafb3932c879a6b8"]},{"id":187743,"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-679681cb5c84444a","e-cafb3932c879a6b8"]},{"id":187744,"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-679681cb5c84444a","e-cafb3932c879a6b8"]},{"id":187745,"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-679681cb5c84444a"]}],"region_exact":{},"incoming_edges":[{"id":"e-679681cb5c84444a","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-e0df8ce9d576df35","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":"df44fbc77c63e4af8396407aacd0f662a3b4abe328608d4ba06e51ef6b5424ac","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-07810086d0e6802d","role":"sequence","name":"basis_map_A013611 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 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-cafb3932c879a6b8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-e0df8ce9d576df35","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":"bf5d82446f169b9ef8b3d01921fb043bdfa5d7cf84192705f075fce1497a5474","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-07810086d0e6802d","role":"sequence","name":"basis_map_A013611 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 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":[],"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:          10,           8;\n  2:         110,         168,           64;\n  3:        1320,        2896,         2112,          512;\n  4:       17160,       48208,        50624,        23552,         4096;\n  5:      240240,      812192,      1094400,       734720,       245760,        32768;\n  6:     3603600,    14104800,     22913536,     19776000,      9564160,      2457600,      262144;\n  7:    57657600,   254505600,    479454976,    499724288,    311234560,    115834880,    23855104,     2097152;\n  8:   980179200,  4787856000,  10186779392,  12330952704,   9288781824,   4459069440,  1332215808,   226492416,   16777216;\n  9: 17643225600, 94022841600, 221664877056, 303451383808, 265845694464, 154573504512, 59652440064, 14734589952, 2113929216, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+9+8*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[(n + 9 + 8*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; for(n = 1, N, my(i = n + 1); v[i] = (n + 9 + 8*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-60c7e36cbe887b7e","steps":["basis A013611","basis A007318"],"edges":["e-40669049634a7e59","e-679681cb5c84444a"],"text":"A137320 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A137320 \u2192 basis A013611 \u2192 basis A007318","chain":null}
