{"id":"n-abad200ec25a15c5","revision":0,"first_index":1,"fingerprint":"f6-5d0768679fdfa04b","name":"shift_x_plus_one of basis_map_A007318 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle 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"],["5","2"],["37","30","6"],["305","372","150","20"],["2641","4300","2610","700","70"],["23525","47910","38850","15680","3150","252"],["213445","521850","529620","285600","86310","13860","924"],["1961825","5597480","6822900","4605720","1859550","449064","60060","3432"],["18205345","59376600","84491820","68514600","34629210","11171160","2246244","257400","12870"],["170195525","624579210","1016225100","962179680","584233650","235927692","63363300","10913760","1093950","48620"],["1600472677","6526806550","11951496810","12940699200","9175336170","4451346900","1496454960","344229600","51853230","4618900","184756"],["15122515985","67844431644","138079162650","168283135020","136461575250","77308751520","31222791600","8989643520","1808299350","242030360","19399380","705432"],["143457011569","702162541860","1572373424622","2130154526500","1944431388900","1259897839200","594196883280","205522574400","51742303470","9247037800","1113524412","81124680","2704156"],["1365435096485","7240695010094","17692142709150","26374506878720","26763393156500","19521622640520","10530371451600","4253253095520","1286324389350","287699271860","46254588380","5062180032","338019500","10400600"],["13033485491077","74435657752050","197071138697160","320588723291200","358000274251620","290304543929400","176288800647960","81439816629600","28761200698230","7727193373900","1554689592456","227149104000","22782514300","1404081000","40116600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","73","847","10321","129367","1651609","21360031","278905249","3668760487","48543499753","645382441711","8614382884849","115367108888311","1549456900170553"],"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":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-5","-2"],["5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["12","10","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-7a3496783bbc03d9","recurrence":{"first_index":1,"initial_polynomials":[["1"],["5","2"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1"]]},"terms":[{"coeff":{"coeffs":[["-2"],["1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["15","6"],["-10","-4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183880,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-3, -2)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"-2"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-3, -2)","region_exact":null,"edges":["e-e65e097d166f9cf4"]},{"id":183881,"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-513a922cc9f00604","e-e65e097d166f9cf4"]},{"id":183882,"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-513a922cc9f00604","e-e65e097d166f9cf4"]},{"id":183883,"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-513a922cc9f00604","e-e65e097d166f9cf4"]},{"id":183884,"position":4,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-513a922cc9f00604"]}],"region_exact":{},"incoming_edges":[{"id":"e-513a922cc9f00604","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-abad200ec25a15c5","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":"a609fd8ee3b3e287d7532790133b3113213d0b30761af0ebde3fef6b9402fec4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0444a4f352052985","role":"sequence","name":"basis_map_A007318 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","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]"}]},{"id":"e-7a3496783bbc03d9","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-abad200ec25a15c5","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":"aedf36876514bc0b0397b0594cb0ebdec9ccfd46f654c336da7f25a336bd132c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-e65e097d166f9cf4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-abad200ec25a15c5","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":"571cdf6a337d2730507ab6f2dc4055e43b3ea57723dd4b18bd42d5729b06f158","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"-2"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0444a4f352052985","role":"sequence","name":"basis_map_A007318 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"-1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-2, -1)"}]}],"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:         5,         2;\n   3:        37,        30,          6;\n   4:       305,       372,        150,        20;\n   5:      2641,      4300,       2610,       700,        70;\n   6:     23525,     47910,      38850,     15680,      3150,       252;\n   7:    213445,    521850,     529620,    285600,     86310,     13860,      924;\n   8:   1961825,   5597480,    6822900,   4605720,   1859550,    449064,    60060,     3432;\n   9:  18205345,  59376600,   84491820,  68514600,  34629210,  11171160,  2246244,   257400,   12870;\n  10: 170195525, 624579210, 1016225100, 962179680, 584233650, 235927692, 63363300, 10913760, 1093950, 48620;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (5*n-5+(2*n-2)*x)*P(n-1,x) + (12+10*x+2*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[((5*n - 5 + (2*n - 2)*x)*P[n - 1, x] + (12 + 10*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];\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] = ((5*n - 5 + (2*n - 2)*x)*v[i - 1] + (12 + 10*x + 2*x^2)*deriv(v[i - 1], x))/(n - 1)); 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-4689fb1386d440ad","steps":["x \u21a6 x+1","basis A007318"],"edges":["e-437a2dd956a9df17","e-513a922cc9f00604"],"text":"A063007 \u2192 x \u21a6 x+1 \u2192 basis A007318"},"headline":"A063007 \u2192 x \u21a6 x+1 \u2192 basis A007318","chain":null}
