{"id":"n-6ec298dee79c9c3c","revision":0,"first_index":1,"fingerprint":"f6-a9fc593edbc03cee","name":"theta_plus_one_squared 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"],["3","8"],["13","72","54"],["63","528","810","320"],["321","3600","8370","6720","1750"],["1683","23640","73710","89600","47250","9072"],["8989","151704","593460","967680","771750","299376","45276"],["48639","958496","4511052","9233280","9875250","5787936","1765764","219648"],["265729","5988384","32942700","81164160","108974250","85621536","39435396","9884160","1042470"],["1462563","37097640","233566740","672967680","1087836750","1072863792","662161500","250398720","53165970","4862000"],["8097453","228297960","1618866810","5340741120","10098338250","11990906928","9287918640","4704860160","1506369150","277134000","22355476"],["45046719","1397469744","11020645350","40964763840","88709370750","123222363264","114908858064","73037352960","31314756330","8683532000","1408394988","101582208"],["251595969","8516894256","73937926062","305755369920","746183938500","1187212650624","1295136098256","990977974272","533024293230","197873676000","48354894588","7009172352","457002364"],["1409933619","51716663544","490095947250","2231718128640","6060493939500","10868271077664","13580978274864","12155939727360","7870105373130","3664542882000","1198544134788","261675767808","34275177300","2038517600"],["7923848253","313062555960","3215725236360","15988825681920","47821488214500","95440014397728","134431456799640","137830222632960","104380837510950","58494950514000","23992478085576","7009172352000","1382432151100","165119925600","9026235000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","139","1721","20761","244955","2838235","32400065","365318785","4076383355","45083885947","494814136217","5394691486393","58471805545067","630482736141547"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-16","-18","39","27","-4"],["16","18","-99","-63","16"],["0","0","80","48","-25"],["0","0","-20","-12","19"],["0","0","0","0","-7"],["0","0","0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-48","-70","-3","33","12","0"],["48","14","-189","-209","-60","0"],["0","56","252","240","51","-8"],["0","0","-60","-56","15","16"],["0","0","0","-8","-21","-10"],["0","0","0","0","3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["16","72","76","-6","-38","-12","0"],["0","0","112","264","200","48","0"],["0","0","-56","-132","-84","0","8"],["0","0","0","0","-16","-24","-8"],["0","0","0","0","4","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183876,"position":0,"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-c7788c463c1d0214"]},{"id":183877,"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-c7788c463c1d0214"]},{"id":183878,"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-c7788c463c1d0214"]},{"id":183879,"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-c7788c463c1d0214"]}],"region_exact":{},"incoming_edges":[{"id":"e-c7788c463c1d0214","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-6ec298dee79c9c3c","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"2d96b58ff758b8b035a26bd891bce8617635fc2027234151dce7929ac58ace88","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]"}]}],"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:       3,        8;\n   3:      13,       72,        54;\n   4:      63,      528,       810,       320;\n   5:     321,     3600,      8370,      6720,       1750;\n   6:    1683,    23640,     73710,     89600,      47250,       9072;\n   7:    8989,   151704,    593460,    967680,     771750,     299376,     45276;\n   8:   48639,   958496,   4511052,   9233280,    9875250,    5787936,   1765764,    219648;\n   9:  265729,  5988384,  32942700,  81164160,  108974250,   85621536,  39435396,   9884160,  1042470;\n  10: 1462563, 37097640, 233566740, 672967680, 1087836750, 1072863792, 662161500, 250398720, 53165970, 4862000;\n","recurrence_sentence":"The row polynomials satisfy (16*n-16+(18*n-18)*x-(20*n^3-80*n^2+99*n-39)*x^2-(12*n^3-48*n^2+63*n-27)*x^3+(n^5-7*n^4+19*n^3-25*n^2+16*n-4)*x^4)*P(n,x) = (48*n-48+(56*n^2+14*n-70)*x-(60*n^3-252*n^2+189*n+3)*x^2-(8*n^4+56*n^3-240*n^2+209*n-33)*x^3+(3*n^5-21*n^4+15*n^3+51*n^2-60*n+12)*x^4+(2*n^5-10*n^4+16*n^3-8*n^2)*x^5)*P(n-1,x) + (16+72*x-(56*n^2-112*n-76)*x^2-(132*n^2-264*n+6)*x^3+(4*n^4-16*n^3-84*n^2+200*n-38)*x^4+(6*n^4-24*n^3+48*n-12)*x^5+(2*n^4-8*n^3+8*n^2)*x^6)*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[((48*n - 48 + (56*n^2 + 14*n - 70)*x - (60*n^3 - 252*n^2 + 189*n + 3)*x^2 - (8*n^4 + 56*n^3 - 240*n^2 + 209*n - 33)*x^3 + (3*n^5 - 21*n^4 + 15*n^3 + 51*n^2 - 60*n + 12)*x^4 + (2*n^5 - 10*n^4 + 16*n^3 - 8*n^2)*x^5)*P[n - 1, x] + (16 + 72*x - (56*n^2 - 112*n - 76)*x^2 - (132*n^2 - 264*n + 6)*x^3 + (4*n^4 - 16*n^3 - 84*n^2 + 200*n - 38)*x^4 + (6*n^4 - 24*n^3 + 48*n - 12)*x^5 + (2*n^4 - 8*n^3 + 8*n^2)*x^6)*D[P[n - 1, x], x])/(16*n - 16 + (18*n - 18)*x - (20*n^3 - 80*n^2 + 99*n - 39)*x^2 - (12*n^3 - 48*n^2 + 63*n - 27)*x^3 + (n^5 - 7*n^4 + 19*n^3 - 25*n^2 + 16*n - 4)*x^4)]];\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] = ((48*n - 48 + (56*n^2 + 14*n - 70)*x - (60*n^3 - 252*n^2 + 189*n + 3)*x^2 - (8*n^4 + 56*n^3 - 240*n^2 + 209*n - 33)*x^3 + (3*n^5 - 21*n^4 + 15*n^3 + 51*n^2 - 60*n + 12)*x^4 + (2*n^5 - 10*n^4 + 16*n^3 - 8*n^2)*x^5)*v[i - 1] + (16 + 72*x - (56*n^2 - 112*n - 76)*x^2 - (132*n^2 - 264*n + 6)*x^3 + (4*n^4 - 16*n^3 - 84*n^2 + 200*n - 38)*x^4 + (6*n^4 - 24*n^3 + 48*n - 12)*x^5 + (2*n^4 - 8*n^3 + 8*n^2)*x^6)*deriv(v[i - 1], x))/(16*n - 16 + (18*n - 18)*x - (20*n^3 - 80*n^2 + 99*n - 39)*x^2 - (12*n^3 - 48*n^2 + 63*n - 27)*x^3 + (n^5 - 7*n^4 + 19*n^3 - 25*n^2 + 16*n - 4)*x^4)); 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","(\u03b8+1)\u00b2"],"edges":["e-437a2dd956a9df17","e-c7788c463c1d0214"],"text":"A063007 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A063007 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}