{"id":"n-8ca1e25abea620be","revision":0,"first_index":0,"fingerprint":"f6-9a9a473010d76dc3","name":"theta_plus_one_squared of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1","4"],["1","12"],["1","20","18"],["1","28","72"],["1","36","162","64"],["1","44","288","320"],["1","52","450","896","200"],["1","60","648","1920","1200"],["1","68","882","3520","4000","576"],["1","76","1152","5824","10000","4032"],["1","84","1458","8960","21000","15552","1568"],["1","92","1800","13056","39200","44352","12544"],["1","100","2178","18240","67200","104832","54880","4096"],["1","108","2592","24640","108000","217728","175616","36864"],["1","116","3042","32384","165000","411264","460992","180224","10368"]],"degrees":[1,1,2,2,3,3,4,4,5,5,6,6,7,7,8],"row_sums":["5","13","39","101","263","653","1599","3829","9047","21085","48623","111045","251527","565549","1263391"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","4"]],"recurrence":{"denominator":{"coeffs":[["-4","32","224","576"],["-8","-4","360","1104"],["-5","-50","144","832"],["-1","-32","-26","308"],["0","-6","-26","56"],["0","0","-4","4"]]},"terms":[{"coeff":{"coeffs":[["-4","16","352","1472","2304"],["-8","-84","256","2344","4992"],["-5","-118","-282","1028","3856"],["-1","-48","-272","-12","1392"],["0","-6","-66","-84","240"],["0","0","-4","-12","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","74","344","-512","-4608"],["0","4","106","712","288","-5376"],["0","1","46","422","652","-2336"],["0","0","6","86","248","-448"],["0","0","0","4","28","-32"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172350,"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-93fe7ce284bae5e9"]},{"id":172351,"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-93fe7ce284bae5e9"]},{"id":172352,"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-93fe7ce284bae5e9"]}],"region_exact":{},"incoming_edges":[{"id":"e-93fe7ce284bae5e9","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-8ca1e25abea620be","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":"354660d422935f6ad515f47867a3701e88a351e91f8d345632fe2b0a184f6aff","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-ff9680a2cfd23e93","role":"sequence","name":"Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/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: 1,  4;\n  1: 1, 12;\n  2: 1, 20,   18;\n  3: 1, 28,   72;\n  4: 1, 36,  162,   64;\n  5: 1, 44,  288,  320;\n  6: 1, 52,  450,  896,   200;\n  7: 1, 60,  648, 1920,  1200;\n  8: 1, 68,  882, 3520,  4000,  576;\n  9: 1, 76, 1152, 5824, 10000, 4032;\n","recurrence_sentence":"The row polynomials satisfy -(n^3+5*n^2+8*n+4+(6*n^4+32*n^3+50*n^2+4*n-32)*x+(4*n^5+26*n^4+26*n^3-144*n^2-360*n-224)*x^2-(4*n^5+56*n^4+308*n^3+832*n^2+1104*n+576)*x^3)*P(n,x) = -(n^3+5*n^2+8*n+4+(6*n^4+48*n^3+118*n^2+84*n-16)*x+(4*n^5+66*n^4+272*n^3+282*n^2-256*n-352)*x^2+(12*n^5+84*n^4+12*n^3-1028*n^2-2344*n-1472)*x^3-(16*n^5+240*n^4+1392*n^3+3856*n^2+4992*n+2304)*x^4)*P(n-1,x) + ((n^2+4*n+4)*x+(6*n^3+46*n^2+106*n+74)*x^2+(4*n^4+86*n^3+422*n^2+712*n+344)*x^3+(28*n^4+248*n^3+652*n^2+288*n-512)*x^4-(32*n^4+448*n^3+2336*n^2+5376*n+4608)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 5*n^2 + 8*n + 4 + (6*n^4 + 48*n^3 + 118*n^2 + 84*n - 16)*x + (4*n^5 + 66*n^4 + 272*n^3 + 282*n^2 - 256*n - 352)*x^2 + (12*n^5 + 84*n^4 + 12*n^3 - 1028*n^2 - 2344*n - 1472)*x^3 - (16*n^5 + 240*n^4 + 1392*n^3 + 3856*n^2 + 4992*n + 2304)*x^4)*P[n - 1, x] + ((n^2 + 4*n + 4)*x + (6*n^3 + 46*n^2 + 106*n + 74)*x^2 + (4*n^4 + 86*n^3 + 422*n^2 + 712*n + 344)*x^3 + (28*n^4 + 248*n^3 + 652*n^2 + 288*n - 512)*x^4 - (32*n^4 + 448*n^3 + 2336*n^2 + 5376*n + 4608)*x^5)*D[P[n - 1, x], x])/(-n^3 - 5*n^2 - 8*n - 4 - (6*n^4 + 32*n^3 + 50*n^2 + 4*n - 32)*x - (4*n^5 + 26*n^4 + 26*n^3 - 144*n^2 - 360*n - 224)*x^2 + (4*n^5 + 56*n^4 + 308*n^3 + 832*n^2 + 1104*n + 576)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 + 5*n^2 + 8*n + 4 + (6*n^4 + 48*n^3 + 118*n^2 + 84*n - 16)*x + (4*n^5 + 66*n^4 + 272*n^3 + 282*n^2 - 256*n - 352)*x^2 + (12*n^5 + 84*n^4 + 12*n^3 - 1028*n^2 - 2344*n - 1472)*x^3 - (16*n^5 + 240*n^4 + 1392*n^3 + 3856*n^2 + 4992*n + 2304)*x^4)*v[i - 1] + ((n^2 + 4*n + 4)*x + (6*n^3 + 46*n^2 + 106*n + 74)*x^2 + (4*n^4 + 86*n^3 + 422*n^2 + 712*n + 344)*x^3 + (28*n^4 + 248*n^3 + 652*n^2 + 288*n - 512)*x^4 - (32*n^4 + 448*n^3 + 2336*n^2 + 5376*n + 4608)*x^5)*deriv(v[i - 1], x))/(-n^3 - 5*n^2 - 8*n - 4 - (6*n^4 + 32*n^3 + 50*n^2 + 4*n - 32)*x - (4*n^5 + 26*n^4 + 26*n^3 - 144*n^2 - 360*n - 224)*x^2 + (4*n^5 + 56*n^4 + 308*n^3 + 832*n^2 + 1104*n + 576)*x^3)); 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-ff9680a2cfd23e93","steps":["(\u03b8+1)\u00b2"],"edges":["e-93fe7ce284bae5e9"],"text":"A199478 \u2192 (\u03b8+1)\u00b2"},"headline":"A199478 \u2192 (\u03b8+1)\u00b2","chain":null}