{"id":"n-ac2000756e296595","revision":0,"first_index":1,"fingerprint":"f6-eec6c58ba17db0cb","name":"theta_plus_one of reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).","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"],["2","2"],["4","10","3"],["8","32","27","4"],["16","88","123","56","5"],["32","224","438","340","100","6"],["64","544","1368","1508","775","162","7"],["128","1280","3936","5632","4165","1554","245","8"],["256","2944","10704","18848","18245","9912","2842","352","9"],["512","6656","27936","58432","70010","50166","21168","4848","486","10"],["1024","14848","70656","171200","244700","218190","122283","41616","7830","650","11"],["2048","32768","174336","480256","797800","852000","597135","271656","76626","12100","847","12"],["4096","71680","421632","1301504","2464480","3068280","2585135","1477520","560439","133760","18029","1080","13"],["8192","155648","1003008","3429376","7291840","10373568","10218670","7045480","3371940","1088230","223366","26052","1352","14"],["16384","335872","2353152","8827904","20827520","33337920","37612400","30402440","17610255","7200050","2008721","359268","36673","1666","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","17","71","288","1140","4428","16948","64112","240224","893008","3297584","12107648","44236736","160930240"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","-6"],["-4","1","16"],["4","-4","-14"],["0","3","4"]]},"terms":[{"coeff":{"coeffs":[["0","0","30","0"],["-8","-38","-39","22"],["8","16","-16","-34"],["0","6","17","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","12","12","0"],["0","-8","-18","-2","8"],["0","0","-6","-14","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174663,"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-428e38212af2fe28"]},{"id":174664,"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-428e38212af2fe28"]},{"id":174665,"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-428e38212af2fe28"]}],"region_exact":{},"incoming_edges":[{"id":"e-428e38212af2fe28","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-ac2000756e296595","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":"e46a05e1b48c85903148e63c45f82b1f7d6462c17a964f30d1944e6b608deb0f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f268708db386b729","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).","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:   2,    2;\n   3:   4,   10,     3;\n   4:   8,   32,    27,     4;\n   5:  16,   88,   123,    56,     5;\n   6:  32,  224,   438,   340,   100,     6;\n   7:  64,  544,  1368,  1508,   775,   162,     7;\n   8: 128, 1280,  3936,  5632,  4165,  1554,   245,    8;\n   9: 256, 2944, 10704, 18848, 18245,  9912,  2842,  352,   9;\n  10: 512, 6656, 27936, 58432, 70010, 50166, 21168, 4848, 486, 10;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-4*n+(3*n^3-4*n^2+n)*x+(4*n^3-14*n^2+16*n-6)*x^2)*P(n,x) = (8*n^2-8*n+(6*n^3+16*n^2-38*n)*x+(17*n^3-16*n^2-39*n+30)*x^2+(12*n^3-34*n^2+22*n)*x^3)*P(n-1,x) - (8*n*x+(6*n^2+18*n-12)*x^2+(14*n^2+2*n-12)*x^3+(8*n^2-8*n)*x^4)*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[((8*n^2 - 8*n + (6*n^3 + 16*n^2 - 38*n)*x + (17*n^3 - 16*n^2 - 39*n + 30)*x^2 + (12*n^3 - 34*n^2 + 22*n)*x^3)*P[n - 1, x] - (8*n*x + (6*n^2 + 18*n - 12)*x^2 + (14*n^2 + 2*n - 12)*x^3 + (8*n^2 - 8*n)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 4*n + (3*n^3 - 4*n^2 + n)*x + (4*n^3 - 14*n^2 + 16*n - 6)*x^2)]];\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] = ((8*n^2 - 8*n + (6*n^3 + 16*n^2 - 38*n)*x + (17*n^3 - 16*n^2 - 39*n + 30)*x^2 + (12*n^3 - 34*n^2 + 22*n)*x^3)*v[i - 1] - (8*n*x + (6*n^2 + 18*n - 12)*x^2 + (14*n^2 + 2*n - 12)*x^3 + (8*n^2 - 8*n)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 4*n + (3*n^3 - 4*n^2 + n)*x + (4*n^3 - 14*n^2 + 16*n - 6)*x^2)); 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-2821c21537bbbcdd","steps":["reverse","\u03b8+1"],"edges":["e-23a5df9ffca078bb","e-428e38212af2fe28"],"text":"A056242 \u2192 reverse \u2192 \u03b8+1"},"headline":"A056242 \u2192 reverse \u2192 \u03b8+1","chain":null}