{"id":"n-d2911c15cf811244","revision":1,"first_index":0,"fingerprint":"f6-097fa7332775e3f8","name":"theta_plus_one_squared of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","4"],["5","20","9"],["14","84","81","16"],["42","336","504","224","25"],["132","1320","2700","1920","500","36"],["429","5148","13365","13200","5625","972","49"],["1430","20020","63063","80080","48125","13860","1715","64"],["4862","77792","288288","448448","350350","144144","30184","2816","81"],["16796","302328","1288872","2376192","2293200","1238328","374556","59904","4374","100"],["58786","1175720","5668650","12093120","13923000","9356256","3745560","873600","110565","6500","121"],["208012","4576264","24622290","59690400","79942500","64465632","32327064","10053120","1871100","192500","9317","144"],["742900","17829600","105907824","287650880","439683750","414421920","250699680","98233344","24538140","3740000","319440","12960","169"],["2674440","69535440","451980360","1359804160","2337163400","2524136472","1792502712","851355648","269374248","55426800","7059624","509184","17576","196"],["9694845","271455660","1916570565","6327725040","12084197125","14724129420","12024705693","6731030592","2592727137","678978300","117366249","12697776","784329","23324","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","34","195","1131","6608","38788","228357","1346965","7954650","47011878","277958343","1643780607","9721540260","57492086280"],"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"]],"recurrence":{"denominator":{"coeffs":[["8","-4","0"],["20","0","0"],["16","19","2"],["4","21","5"],["0","6","4"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["8","-4","0","0"],["32","42","0","2"],["40","118","58","9"],["16","96","90","20"],["0","24","40","14"],["0","0","4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-8","-14","-8","-2"],["0","-16","-47","-40","-9"],["0","-8","-44","-52","-16"],["0","0","-12","-22","-10"],["0","0","0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164904,"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-e07ba370ee736991"]},{"id":164905,"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-e07ba370ee736991"]},{"id":164906,"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-e07ba370ee736991"]},{"id":164907,"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-e07ba370ee736991"]}],"region_exact":{},"incoming_edges":[{"id":"e-e07ba370ee736991","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-d2911c15cf811244","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":"25b746d9fbf3cd3e1e86d4b28415f46867223af9bd5c7d035b6dbb91ec63ed28","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-57b4b14604f4c0d7","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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;\n  1:     2,      4;\n  2:     5,     20,       9;\n  3:    14,     84,      81,      16;\n  4:    42,    336,     504,     224,      25;\n  5:   132,   1320,    2700,    1920,     500,      36;\n  6:   429,   5148,   13365,   13200,    5625,     972,     49;\n  7:  1430,  20020,   63063,   80080,   48125,   13860,   1715,    64;\n  8:  4862,  77792,  288288,  448448,  350350,  144144,  30184,  2816,   81;\n  9: 16796, 302328, 1288872, 2376192, 2293200, 1238328, 374556, 59904, 4374, 100;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3+16*n^2+20*n+8+(6*n^4+21*n^3+19*n^2-4)*x+(n^5+4*n^4+5*n^3+2*n^2)*x^2)*P(n,x) = (16*n^3+40*n^2+32*n+8+(24*n^4+96*n^3+118*n^2+42*n-4)*x+(4*n^5+40*n^4+90*n^3+58*n^2)*x^2+(3*n^5+14*n^4+20*n^3+9*n^2+2*n)*x^3)*P(n-1,x) - ((8*n^2+16*n+8)*x+(12*n^3+44*n^2+47*n+14)*x^2+(2*n^4+22*n^3+52*n^2+40*n+8)*x^3+(2*n^4+10*n^3+16*n^2+9*n+2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 + 40*n^2 + 32*n + 8 + (24*n^4 + 96*n^3 + 118*n^2 + 42*n - 4)*x + (4*n^5 + 40*n^4 + 90*n^3 + 58*n^2)*x^2 + (3*n^5 + 14*n^4 + 20*n^3 + 9*n^2 + 2*n)*x^3)*P[n - 1, x] - ((8*n^2 + 16*n + 8)*x + (12*n^3 + 44*n^2 + 47*n + 14)*x^2 + (2*n^4 + 22*n^3 + 52*n^2 + 40*n + 8)*x^3 + (2*n^4 + 10*n^3 + 16*n^2 + 9*n + 2)*x^4)*D[P[n - 1, x], x])/(4*n^3 + 16*n^2 + 20*n + 8 + (6*n^4 + 21*n^3 + 19*n^2 - 4)*x + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 + 40*n^2 + 32*n + 8 + (24*n^4 + 96*n^3 + 118*n^2 + 42*n - 4)*x + (4*n^5 + 40*n^4 + 90*n^3 + 58*n^2)*x^2 + (3*n^5 + 14*n^4 + 20*n^3 + 9*n^2 + 2*n)*x^3)*v[i - 1] - ((8*n^2 + 16*n + 8)*x + (12*n^3 + 44*n^2 + 47*n + 14)*x^2 + (2*n^4 + 22*n^3 + 52*n^2 + 40*n + 8)*x^3 + (2*n^4 + 10*n^3 + 16*n^2 + 9*n + 2)*x^4)*deriv(v[i - 1], x))/(4*n^3 + 16*n^2 + 20*n + 8 + (6*n^4 + 21*n^3 + 19*n^2 - 4)*x + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*x^2)); 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-57b4b14604f4c0d7","steps":["(\u03b8+1)\u00b2"],"edges":["e-e07ba370ee736991"],"text":"A126216 \u2192 (\u03b8+1)\u00b2"},"headline":"A126216 \u2192 (\u03b8+1)\u00b2","chain":null}