{"id":"n-8a2aac8e257e6387","revision":0,"first_index":0,"fingerprint":"f6-d6df0c9faac026eb","name":"one_plus_derivative of reversal of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["6","2"],["20","20","4"],["64","136","80","12"],["192","720","880","392","48"],["544","3200","6760","6184","2296","240"],["1472","12544","40880","63728","48272","15696","1440"],["3840","44800","208768","495936","633584","418688","122832","10080"],["9728","148992","940800","3163776","6101088","6746496","4013344","1083456","80640"],["24064","468480","3851520","17407488","47141472","78091680","77250080","42232032","10638144","725760"],["58368","1408000","14615040","85426176","308852544","713782080","1051205760","950780864","484659648","115090560","7257600"],["139264","4077568","52152320","382728192","1779537408","5459988864","11156773760","14944897088","12550223488","6029097408","1360344960","79833600"],["327680","11448320","176889856","1592008704","9249816576","36325715712","98078176768","181433439616","224648762624","177149002368","80855963136","17441809920","958003200"],["761856","31309824","574945280","6225995776","44184172032","215950884864","743254402176","1809333542016","3081330284032","3568847015552","2665531150464","1163306170368","241113576960","12454041600"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","2","8","44","292","2232","19224","184032","1938528","22288320","277830720","3733136640","53799793920","827921364480","13552138252800"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"],["6","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","-8","4"],["-4","12","-8"],["0","-4","5"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-6","4"],["4","-6"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177412,"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-e328c92b880f32de"]},{"id":177413,"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-e328c92b880f32de"]},{"id":177414,"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-e328c92b880f32de"]},{"id":177415,"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-e328c92b880f32de"]}],"region_exact":{},"incoming_edges":[{"id":"e-e328c92b880f32de","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-8a2aac8e257e6387","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"744d639bd785cc1db7660bcd616127d3b7711ad4cffff7bb8b2213c2f08ad7df","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-3df3926aefc1fd76","role":"sequence","name":"reversal of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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;\n  2:    6,      2;\n  3:   20,     20,      4;\n  4:   64,    136,     80,      12;\n  5:  192,    720,    880,     392,      48;\n  6:  544,   3200,   6760,    6184,    2296,     240;\n  7: 1472,  12544,  40880,   63728,   48272,   15696,    1440;\n  8: 3840,  44800, 208768,  495936,  633584,  418688,  122832,   10080;\n  9: 9728, 148992, 940800, 3163776, 6101088, 6746496, 4013344, 1083456, 80640;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(4*n-4+(4*n^2-12*n+8)*x+(n^3-5*n^2+8*n-4)*x^2)*P(n-2,x) + (4*n-6+(2*n^2-6*n+4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 6 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 6 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 4 + (4*n^2 - 12*n + 8)*x + (n^3 - 5*n^2 + 8*n - 4)*x^2)*P[n - 2, x] + (4*n - 6 + (2*n^2 - 6*n + 4)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 6 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 4 + (4*n^2 - 12*n + 8)*x + (n^3 - 5*n^2 + 8*n - 4)*x^2)*v[i - 2] + (4*n - 6 + (2*n^2 - 6*n + 4)*x)*v[i - 1])/(n - 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-60c7e36cbe887b7e","steps":["reverse","D+1"],"edges":["e-a28ae8b138b0cf1e","e-e328c92b880f32de"],"text":"A137320 \u2192 reverse \u2192 D+1"},"headline":"A137320 \u2192 reverse \u2192 D+1","chain":null}