{"id":"n-2869b22ec20dd5b7","revision":0,"first_index":1,"fingerprint":"f6-9aff797b8fdd9c6c","name":"theta_plus_one_squared of basis_map_A013610 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","24"],["13","216","486"],["63","1584","7290","8640"],["321","10800","75330","181440","141750"],["1683","70920","663390","2419200","3827250","2204496"],["8989","455112","5341140","26127360","62511750","72748368","33006204"],["48639","2875488","40599468","249298560","799895250","1406468448","1287241956","480370176"],["265729","17965152","296484300","2191432320","8826914250","20806033248","28748403684","21616657920","6839645670"],["1462563","111292920","2102100660","18170127360","88114776750","260705901456","482715733500","547622000640","348821929170","95698746000"],["8097453","684893880","14569801290","144200010240","817965398250","2913790383504","6770892688560","10289529169920","9883287993150","5454828522000","1320068502324"],["45046719","4192409232","99185808150","1106048623680","7185459030750","29943034273152","83768557528656","159732690923520","205456116281130","170917960356000","83164315646412","17994983400576"],["251595969","25550682768","665441334558","8255394987840","60440899018500","288492674101632","944154215628624","2167268829732864","3497172387882030","3894747564708000","2855308170526812","1241653854639744","242869793326524"],["1409933619","155149990632","4410863525250","60256389473280","490900009099500","2640989871872352","9900533162375856","26585040183736320","51635761353105930","72129197546406000","70772832615096612","46355077239883776","18215234499489300","3250055495584800"],["7923848253","939187667880","28941527127240","431698293411840","3873540545374500","23191923498647904","98000532006937560","301434696898283520","684842674909342950","1151356110967062000","1416731838475177224","1241653854639744000","734681124812735100","263254495142368800","43172202191715000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","27","715","17577","409641","9186939","200198923","4266797985","89343802273","1844064071019","37609825460571","759372589327977","15201055028165865","302040445789573227","5962654581019443771"],"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":[["-48","-162","1053","2187","-972"],["48","162","-2673","-5103","3888"],["0","0","2160","3888","-6075"],["0","0","-540","-972","4617"],["0","0","0","0","-1701"],["0","0","0","0","243"]]},"terms":[{"coeff":{"coeffs":[["-144","-630","-81","2673","2916","0"],["144","126","-5103","-16929","-14580","0"],["0","504","6804","19440","12393","-5832"],["0","0","-1620","-4536","3645","11664"],["0","0","0","-648","-5103","-7290"],["0","0","0","0","729","1458"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["16","216","684","-162","-3078","-2916","0"],["0","0","1008","7128","16200","11664","0"],["0","0","-504","-3564","-6804","0","5832"],["0","0","0","0","-1296","-5832","-5832"],["0","0","0","0","324","1458","1458"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188936,"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-d85489cf1e46e0df"]},{"id":188937,"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-d85489cf1e46e0df"]},{"id":188938,"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-d85489cf1e46e0df"]},{"id":188939,"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-d85489cf1e46e0df"]}],"region_exact":{},"incoming_edges":[{"id":"e-d85489cf1e46e0df","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-2869b22ec20dd5b7","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":"850bdac4049b651ebce85c0584a923afa7c8e848bea08bd6f31a027091b2570f","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-ecc015942a4103af","role":"sequence","name":"basis_map_A013610 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,        24;\n   3:      13,       216,        486;\n   4:      63,      1584,       7290,        8640;\n   5:     321,     10800,      75330,      181440,      141750;\n   6:    1683,     70920,     663390,     2419200,     3827250,      2204496;\n   7:    8989,    455112,    5341140,    26127360,    62511750,     72748368,     33006204;\n   8:   48639,   2875488,   40599468,   249298560,   799895250,   1406468448,   1287241956,    480370176;\n   9:  265729,  17965152,  296484300,  2191432320,  8826914250,  20806033248,  28748403684,  21616657920,   6839645670;\n  10: 1462563, 111292920, 2102100660, 18170127360, 88114776750, 260705901456, 482715733500, 547622000640, 348821929170, 95698746000;\n","recurrence_sentence":"The row polynomials satisfy (48*n-48+(162*n-162)*x-(540*n^3-2160*n^2+2673*n-1053)*x^2-(972*n^3-3888*n^2+5103*n-2187)*x^3+(243*n^5-1701*n^4+4617*n^3-6075*n^2+3888*n-972)*x^4)*P(n,x) = (144*n-144+(504*n^2+126*n-630)*x-(1620*n^3-6804*n^2+5103*n+81)*x^2-(648*n^4+4536*n^3-19440*n^2+16929*n-2673)*x^3+(729*n^5-5103*n^4+3645*n^3+12393*n^2-14580*n+2916)*x^4+(1458*n^5-7290*n^4+11664*n^3-5832*n^2)*x^5)*P(n-1,x) + (16+216*x-(504*n^2-1008*n-684)*x^2-(3564*n^2-7128*n+162)*x^3+(324*n^4-1296*n^3-6804*n^2+16200*n-3078)*x^4+(1458*n^4-5832*n^3+11664*n-2916)*x^5+(1458*n^4-5832*n^3+5832*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[((144*n - 144 + (504*n^2 + 126*n - 630)*x - (1620*n^3 - 6804*n^2 + 5103*n + 81)*x^2 - (648*n^4 + 4536*n^3 - 19440*n^2 + 16929*n - 2673)*x^3 + (729*n^5 - 5103*n^4 + 3645*n^3 + 12393*n^2 - 14580*n + 2916)*x^4 + (1458*n^5 - 7290*n^4 + 11664*n^3 - 5832*n^2)*x^5)*P[n - 1, x] + (16 + 216*x - (504*n^2 - 1008*n - 684)*x^2 - (3564*n^2 - 7128*n + 162)*x^3 + (324*n^4 - 1296*n^3 - 6804*n^2 + 16200*n - 3078)*x^4 + (1458*n^4 - 5832*n^3 + 11664*n - 2916)*x^5 + (1458*n^4 - 5832*n^3 + 5832*n^2)*x^6)*D[P[n - 1, x], x])/(48*n - 48 + (162*n - 162)*x - (540*n^3 - 2160*n^2 + 2673*n - 1053)*x^2 - (972*n^3 - 3888*n^2 + 5103*n - 2187)*x^3 + (243*n^5 - 1701*n^4 + 4617*n^3 - 6075*n^2 + 3888*n - 972)*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] = ((144*n - 144 + (504*n^2 + 126*n - 630)*x - (1620*n^3 - 6804*n^2 + 5103*n + 81)*x^2 - (648*n^4 + 4536*n^3 - 19440*n^2 + 16929*n - 2673)*x^3 + (729*n^5 - 5103*n^4 + 3645*n^3 + 12393*n^2 - 14580*n + 2916)*x^4 + (1458*n^5 - 7290*n^4 + 11664*n^3 - 5832*n^2)*x^5)*v[i - 1] + (16 + 216*x - (504*n^2 - 1008*n - 684)*x^2 - (3564*n^2 - 7128*n + 162)*x^3 + (324*n^4 - 1296*n^3 - 6804*n^2 + 16200*n - 3078)*x^4 + (1458*n^4 - 5832*n^3 + 11664*n - 2916)*x^5 + (1458*n^4 - 5832*n^3 + 5832*n^2)*x^6)*deriv(v[i - 1], x))/(48*n - 48 + (162*n - 162)*x - (540*n^3 - 2160*n^2 + 2673*n - 1053)*x^2 - (972*n^3 - 3888*n^2 + 5103*n - 2187)*x^3 + (243*n^5 - 1701*n^4 + 4617*n^3 - 6075*n^2 + 3888*n - 972)*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":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-50c3f3c57fe6951a","e-d85489cf1e46e0df"],"text":"A063007 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A063007 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}