{"id":"n-cd2d00ce01d1d8bf","revision":1,"first_index":0,"fingerprint":"f6-b6fea4e2006d7377","name":"theta_plus_one_squared 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":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["4","16","9"],["4","32","45","16"],["4","48","117","96","25"],["4","64","225","304","175","36"],["4","80","369","704","650","288","49"],["4","96","549","1360","1750","1224","441","64"],["4","112","765","2336","3875","3744","2107","640","81"],["4","128","1017","3696","7525","9324","7203","3392","891","100"],["4","144","1305","5504","13300","20160","19894","12800","5184","1200","121"],["4","160","1629","7824","21900","39312","47334","38784","21384","7600","1573","144"],["4","176","1989","10720","34125","70848","100842","100608","70470","34000","10769","2016","169"],["4","192","2385","14256","50875","119988","197274","232320","197802","121000","51909","14832","2535","196"],["4","208","2817","18496","73150","193248","360591","489984","491832","365200","198319","76608","19942","3136","225"],["4","224","3285","23504","102050","298584","623623","960960","1111968","972400","640211","312624","109850","26264","3825","256"],["4","240","3789","29344","138775","445536","1030029","1775488","2328183","2345200","1816815","1074528","476749","153664","33975","4608","289"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["29","97","290","808","2144","5488","13664","33280","79616","187648","436736","1005568","2293760","5189632","11657216"],"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","16","9"]],"recurrence":{"denominator":{"coeffs":[["4","12","21","15","4"],["0","12","20","15","4"],["0","0","4","4","1"]]},"terms":[{"coeff":{"coeffs":[["4","28","69","79","43","9"],["0","12","40","51","29","6"],["0","0","4","8","5","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163430,"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-247d5155af4042c4"]},{"id":163431,"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-247d5155af4042c4"]},{"id":163432,"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-247d5155af4042c4"]},{"id":163433,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-247d5155af4042c4"]}],"region_exact":{},"incoming_edges":[{"id":"e-247d5155af4042c4","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-cd2d00ce01d1d8bf","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":"516f47a7bb2b1c26792f060f4801d46850ad3cbd3f5bf15a9a19ae8ee36f14cd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f48bed27dbc29c5f","role":"sequence","name":"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,  16,    9;\n  1: 4,  32,   45,   16;\n  2: 4,  48,  117,   96,    25;\n  3: 4,  64,  225,  304,   175,    36;\n  4: 4,  80,  369,  704,   650,   288,    49;\n  5: 4,  96,  549, 1360,  1750,  1224,   441,    64;\n  6: 4, 112,  765, 2336,  3875,  3744,  2107,   640,    81;\n  7: 4, 128, 1017, 3696,  7525,  9324,  7203,  3392,   891,  100;\n  8: 4, 144, 1305, 5504, 13300, 20160, 19894, 12800,  5184, 1200,  121;\n  9: 4, 160, 1629, 7824, 21900, 39312, 47334, 38784, 21384, 7600, 1573, 144;\n","recurrence_sentence":"The row polynomials satisfy (4+(12*n+12)*x+(4*n^2+20*n+21)*x^2+(4*n^2+15*n+15)*x^3+(n^2+4*n+4)*x^4)*P(n,x) = (4+(12*n+28)*x+(4*n^2+40*n+69)*x^2+(8*n^2+51*n+79)*x^3+(5*n^2+29*n+43)*x^4+(n^2+6*n+9)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 16*x + 9*x^2.","mathematica":"P[0, x_] := 4 + 16*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4 + (12*n + 28)*x + (4*n^2 + 40*n + 69)*x^2 + (8*n^2 + 51*n + 79)*x^3 + (5*n^2 + 29*n + 43)*x^4 + (n^2 + 6*n + 9)*x^5)*P[n - 1, x])/(4 + (12*n + 12)*x + (4*n^2 + 20*n + 21)*x^2 + (4*n^2 + 15*n + 15)*x^3 + (n^2 + 4*n + 4)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4 + 16*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4 + (12*n + 28)*x + (4*n^2 + 40*n + 69)*x^2 + (8*n^2 + 51*n + 79)*x^3 + (5*n^2 + 29*n + 43)*x^4 + (n^2 + 6*n + 9)*x^5)*v[i - 1])/(4 + (12*n + 12)*x + (4*n^2 + 20*n + 21)*x^2 + (4*n^2 + 15*n + 15)*x^3 + (n^2 + 4*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f48bed27dbc29c5f","steps":["(\u03b8+1)\u00b2"],"edges":["e-247d5155af4042c4"],"text":"A391533 \u2192 (\u03b8+1)\u00b2"},"headline":"A391533 \u2192 (\u03b8+1)\u00b2","chain":null}
