{"id":"n-beacdfa3ad6b4129","revision":0,"first_index":0,"fingerprint":"f6-75cc7c95cefa889b","name":"theta_plus_one_squared of veronese_section_2_1 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["4"],["8","4"],["12","24"],["16","76","9"],["20","176","72"],["24","340","306","16"],["28","584","936","160"],["32","924","2331","848","25"],["36","1376","5040","3200","300"],["40","1956","9828","9696","1900","36"],["44","2680","17712","25152","8500","504"],["48","3564","29997","58080","30250","3708","49"],["52","4624","48312","122496","91300","19152","784"],["56","5876","74646","240240","243100","78156","6566","64"],["60","7336","111384","443872","586300","268632","38416","1152"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["4","12","36","101","268","686","1708","4160","9952","23456","54592","125696","286720","648704","1457152"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4"]],"recurrence":{"denominator":{"coeffs":[["0","-192","-96","-60","12"],["64","48","296","-138","40"],["0","-32","296","-144","51"],["0","-160","80","-102","31"],["0","0","32","-44","9"],["0","0","16","-8","1"]]},"terms":[{"coeff":{"coeffs":[["64","-528","-592","-384","48","0"],["64","272","1416","-60","-4","36"],["0","-192","200","38","-88","60"],["0","-160","-176","10","-109","37"],["0","0","48","20","-48","10"],["0","0","16","8","-7","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","128","-96","480","-792","352","-72"],["0","0","-704","1376","-1180","628","-120"],["0","0","-320","800","-860","454","-74"],["0","0","0","192","-320","148","-20"],["0","0","0","32","-48","18","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173079,"position":0,"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-7523c30b1846fc7b"]},{"id":173080,"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-7523c30b1846fc7b"]},{"id":173081,"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-7523c30b1846fc7b"]}],"region_exact":{},"incoming_edges":[{"id":"e-7523c30b1846fc7b","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-beacdfa3ad6b4129","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":"27902364066fcbb856fe55821412e3a704961f9aaaae05eb2a1d5b5db68018a3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6feee566fc77ea06","role":"sequence","name":"veronese_section_2_1 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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:  4;\n  1:  8,    4;\n  2: 12,   24;\n  3: 16,   76,    9;\n  4: 20,  176,   72;\n  5: 24,  340,  306,   16;\n  6: 28,  584,  936,  160;\n  7: 32,  924, 2331,  848,   25;\n  8: 36, 1376, 5040, 3200,  300;\n  9: 40, 1956, 9828, 9696, 1900, 36;\n","recurrence_sentence":"The row polynomials satisfy (64*n-(160*n^3+32*n^2-48*n+192)*x+(16*n^5+32*n^4+80*n^3+296*n^2+296*n-96)*x^2-(8*n^5+44*n^4+102*n^3+144*n^2+138*n+60)*x^3+(n^5+9*n^4+31*n^3+51*n^2+40*n+12)*x^4)*P(n,x) = (64*n+64-(160*n^3+192*n^2-272*n+528)*x+(16*n^5+48*n^4-176*n^3+200*n^2+1416*n-592)*x^2+(8*n^5+20*n^4+10*n^3+38*n^2-60*n-384)*x^3-(7*n^5+48*n^4+109*n^3+88*n^2+4*n-48)*x^4+(n^5+10*n^4+37*n^3+60*n^2+36*n)*x^5)*P(n-1,x) + (128*x-(320*n^2+704*n+96)*x^2+(32*n^4+192*n^3+800*n^2+1376*n+480)*x^3-(48*n^4+320*n^3+860*n^2+1180*n+792)*x^4+(18*n^4+148*n^3+454*n^2+628*n+352)*x^5-(2*n^4+20*n^3+74*n^2+120*n+72)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4.","mathematica":"P[0, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((64*n + 64 - (160*n^3 + 192*n^2 - 272*n + 528)*x + (16*n^5 + 48*n^4 - 176*n^3 + 200*n^2 + 1416*n - 592)*x^2 + (8*n^5 + 20*n^4 + 10*n^3 + 38*n^2 - 60*n - 384)*x^3 - (7*n^5 + 48*n^4 + 109*n^3 + 88*n^2 + 4*n - 48)*x^4 + (n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n)*x^5)*P[n - 1, x] + (128*x - (320*n^2 + 704*n + 96)*x^2 + (32*n^4 + 192*n^3 + 800*n^2 + 1376*n + 480)*x^3 - (48*n^4 + 320*n^3 + 860*n^2 + 1180*n + 792)*x^4 + (18*n^4 + 148*n^3 + 454*n^2 + 628*n + 352)*x^5 - (2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x^6)*D[P[n - 1, x], x])/(64*n - (160*n^3 + 32*n^2 - 48*n + 192)*x + (16*n^5 + 32*n^4 + 80*n^3 + 296*n^2 + 296*n - 96)*x^2 - (8*n^5 + 44*n^4 + 102*n^3 + 144*n^2 + 138*n + 60)*x^3 + (n^5 + 9*n^4 + 31*n^3 + 51*n^2 + 40*n + 12)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4; for(n = 1, N, my(i = n + 1); v[i] = ((64*n + 64 - (160*n^3 + 192*n^2 - 272*n + 528)*x + (16*n^5 + 48*n^4 - 176*n^3 + 200*n^2 + 1416*n - 592)*x^2 + (8*n^5 + 20*n^4 + 10*n^3 + 38*n^2 - 60*n - 384)*x^3 - (7*n^5 + 48*n^4 + 109*n^3 + 88*n^2 + 4*n - 48)*x^4 + (n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n)*x^5)*v[i - 1] + (128*x - (320*n^2 + 704*n + 96)*x^2 + (32*n^4 + 192*n^3 + 800*n^2 + 1376*n + 480)*x^3 - (48*n^4 + 320*n^3 + 860*n^2 + 1180*n + 792)*x^4 + (18*n^4 + 148*n^3 + 454*n^2 + 628*n + 352)*x^5 - (2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x^6)*deriv(v[i - 1], x))/(64*n - (160*n^3 + 32*n^2 - 48*n + 192)*x + (16*n^5 + 32*n^4 + 80*n^3 + 296*n^2 + 296*n - 96)*x^2 - (8*n^5 + 44*n^4 + 102*n^3 + 144*n^2 + 138*n + 60)*x^3 + (n^5 + 9*n^4 + 31*n^3 + 51*n^2 + 40*n + 12)*x^4)); 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-f48bed27dbc29c5f","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-a61d9f5d8a4277bd","e-7523c30b1846fc7b"],"text":"A391533 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A391533 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}