{"id":"n-8c995547482f9480","revision":0,"first_index":1,"fingerprint":"f6-4748f217d058d591","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.","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":[["2"],["16","8"],["96","96","20"],["512","768","336","40"],["2560","5120","3456","896","70"],["12288","30720","28160","11520","2016","112"],["57344","172032","199680","112640","31680","4032","168"],["262144","917504","1290240","931840","366080","76032","7392","240"],["1179648","4718592","7798784","6881280","3494400","1025024","164736","12672","330"],["5242880","23592960","44826624","46792704","29245440","11182080","2562560","329472","20592","440"],["23068672","115343360","247726080","298844160","222265344","105283584","31682560","5857280","617760","32032","572"],["100663296","553648128","1326448640","1816657920","1568931840","889061376","333398016","81469440","12446720","1098240","48048","728"],["436207616","2617245696","6920601600","10611589120","10445783040","6903300096","3111714816","952565760","193489920","24893440","1867008","69888","910"],["1879048192","12213813248","35332816896","59978547200","66322432000","50139758592","26462650368","9779675136","2500485120","429977600","47297536","3055104","99008","1120"],["8053063680","56371445760","177100292096","329772957696","404855193600","344876646400","208915660800","90729086976","28116566016","6112296960","902952960","85995520","4837248","137088","1360"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","24","212","1656","12102","84816","577576","3851472","25275466","163795752","1050721404","6683872392","42219328910","265089657120","1655897134160"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-2"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179563,"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-ccf60822171bc8ef"]},{"id":179564,"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-ccf60822171bc8ef"]},{"id":179565,"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-ccf60822171bc8ef"]},{"id":179566,"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-ccf60822171bc8ef"]}],"region_exact":{},"incoming_edges":[{"id":"e-ccf60822171bc8ef","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-8c995547482f9480","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":"b1b2b0b5f8d0a80fc1ce92e7a9d6e28a3950eb2da39fe99fd9c4a791237456fa","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-92be85f47bd42a26","role":"sequence","name":"Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.","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-7091d188d4739a26","kind":"fingerprint_related","node_a_id":"n-8c995547482f9480","node_b_id":"n-dbf55b6b36baa8f7","fingerprint":"f6-4748f217d058d591","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4748f217d058d591"},"other_id":"n-dbf55b6b36baa8f7","other":{"id":"n-dbf55b6b36baa8f7","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.","first_index":1,"fingerprint":"f6-4748f217d058d591","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["8","16"],["20","96","96"]],"card_rows":[["2"],["8","16"],["20","96","96"],["40","336","768","512"]],"row_sums_preview":["2","24","212","1656","12102","84816","577576","3851472"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-92be85f47bd42a26","steps":["x \u21a6 x/(x+1)"],"edges":["e-947702eb73f76d5c"],"text":"A375853 \u2192 x \u21a6 x/(x+1)"},"headline":"A375853 \u2192 x \u21a6 x/(x+1)","chain":null}}],"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:       2;\n   2:      16,        8;\n   3:      96,       96,       20;\n   4:     512,      768,      336,       40;\n   5:    2560,     5120,     3456,      896,       70;\n   6:   12288,    30720,    28160,    11520,     2016,      112;\n   7:   57344,   172032,   199680,   112640,    31680,     4032,     168;\n   8:  262144,   917504,  1290240,   931840,   366080,    76032,    7392,    240;\n   9: 1179648,  4718592,  7798784,  6881280,  3494400,  1025024,  164736,  12672,   330;\n  10: 5242880, 23592960, 44826624, 46792704, 29245440, 11182080, 2562560, 329472, 20592, 440;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (4*n+(3*n-2)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.","mathematica":"P[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n + (3*n - 2)*x)*P[n - 1, x] - (2*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] = 2; for(n = 2, N, my(i = n); v[i] = ((4*n + (3*n - 2)*x)*v[i - 1] - (2*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-92be85f47bd42a26","steps":["x \u21a6 x+1"],"edges":["e-ccf60822171bc8ef"],"text":"A375853 \u2192 x \u21a6 x+1"},"headline":"A375853 \u2192 x \u21a6 x+1","chain":null}